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A000079 - OEIS

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A000079 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A000079" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA000079">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A000079 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A000079 </div> <div class=seqname> Powers of 2: a(n) = 2^n. <br><font size=-1>(Formerly M1129 N0432)</font> </div> </div> <div class=scorerefs> 3235 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592</div> <div class=seqdatalinks> (<a href="/A000079/list">list</a>; <a href="/A000079/graph">graph</a>; <a href="/search?q=A000079+-id:A000079">refs</a>; <a href="/A000079/listen">listen</a>; <a href="/history?seq=A000079">history</a>; <a href="/search?q=id:A000079&fmt=text">text</a>; <a href="/A000079/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>2^0 = 1 is the only odd power of 2.</div> <div class=sectline>Number of subsets of an n-set.</div> <div class=sectline>There are 2^(n-1) compositions (ordered partitions) of n (see for example Riordan). This is the unlabeled analog of the preferential labelings sequence <a href="/A000670" title="Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; o...">A000670</a>.</div> <div class=sectline>This is also the number of weakly unimodal permutations of 1..n + 1, that is, permutations with exactly one local maximum. E.g., a(4) = 16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - <a href="/wiki/User:Jon_Perry">Jon Perry</a>, Jul 27 2003 [Proof: see next line! See also <a href="/A087783" title="Array T(n,k) (n &gt;= 1, k &gt;= 1) read by antidiagonals, giving number of ways of arranging the numbers 1 ... mn into an m X n a...">A087783</a>.]</div> <div class=sectline>Proof: n must appear somewhere and there are 2^(n-1) possible choices for the subset that precedes it. These must appear in increasing order and the rest must follow n in decreasing order. QED. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Oct 26 2003</div> <div class=sectline>a(n+1) is the smallest number that is not the sum of any number of (distinct) earlier terms.</div> <div class=sectline>Same as Pisot sequences E(1, 2), L(1, 2), P(1, 2), T(1, 2). See <a href="/A008776" title="Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).">A008776</a> for definitions of Pisot sequences.</div> <div class=sectline>With initial 1 omitted, same as Pisot sequences E(2, 4), L(2, 4), P(2, 4), T(2, 4). - <a href="/wiki/User:David_W._Wilson">David W. Wilson</a></div> <div class=sectline>Not the sum of two or more consecutive numbers. - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, May 14 2004</div> <div class=sectline>Least deficient or near-perfect numbers (i.e., n such that sigma(n) = <a href="/A000203" title="a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).">A000203</a>(n) = 2n - 1). - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, Jun 03 2004. [Comment from <a href="/wiki/User:Max_Alekseyev">Max Alekseyev</a>, Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2.]</div> <div class=sectline>Almost-perfect numbers referred to as least deficient or slightly defective (Singh 1997) numbers. Does &quot;near-perfect numbers&quot; refer to both almost-perfect numbers (sigma(n) = 2n - 1) and quasi-perfect numbers (sigma(n) = 2n + 1)? There are no known quasi-perfect or least abundant or slightly excessive (Singh 1997) numbers.</div> <div class=sectline>The sum of the numbers in the n-th row of Pascal's triangle; the sum of the coefficients of x in the expansion of (x+1)^n.</div> <div class=sectline>The Collatz conjecture (the hailstone sequence will eventually reach the number 1, regardless of which positive integer is chosen initially) may be restated as (the hailstone sequence will eventually reach a power of 2, regardless of which positive integer is chosen initially).</div> <div class=sectline>The only hailstone sequence which doesn't rebound (except &quot;on the ground&quot;). - <a href="/wiki/User:Alexandre_Wajnberg">Alexandre Wajnberg</a>, Jan 29 2005</div> <div class=sectline>With p(n) as the number of integer partitions of n, p(i) is the number of parts of the i-th partition of n, d(i) is the number of different parts of the i-th partition of n, m(i,j) is the multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i = 1..p(n)} (p(i)! / (Product_{j=1..d(i)} m(i,j)!)). - <a href="/wiki/User:Thomas_Wieder">Thomas Wieder</a>, May 18 2005</div> <div class=sectline>The number of binary relations on an n-element set that are both symmetric and antisymmetric. Also the number of binary relations on an n-element set that are symmetric, antisymmetric and transitive.</div> <div class=sectline>The first differences are the sequence itself. - <a href="/wiki/User:Alexandre_Wajnberg">Alexandre Wajnberg</a> and <a href="/wiki/User:Eric_Angelini">Eric Angelini</a>, Sep 07 2005</div> <div class=sectline>a(n) is the largest number with shortest addition chain involving n additions. - <a href="/wiki/User:David_W._Wilson">David W. Wilson</a>, Apr 23 2006</div> <div class=sectline>Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - <a href="/wiki/User:Giovanni_Teofilatto">Giovanni Teofilatto</a>, Aug 06 2006</div> <div class=sectline>For n &gt;= 1, a(n) is equal to the number of functions f:{1, 2, ..., n} -&gt; {1, 2} such that for a fixed x in {1, 2, ..., n} and a fixed y in {1, 2} we have f(x) != y. - Aleksandar M. Janjic and <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, Mar 27 2007</div> <div class=sectline>Let P(A) be the power set of an n-element set A. Then a(n) is the number of pairs of elements {x,y} of P(A) for which x = y. - <a href="/wiki/User:Ross_La_Haye">Ross La Haye</a>, Jan 09 2008</div> <div class=sectline>a(n) is the number of different ways to run up a staircase with n steps, taking steps of sizes 1, 2, 3, ... and r (r &lt;= n), where the order IS important and there is no restriction on the number or the size of each step taken. - <a href="/wiki/User:Mohammad_K._Azarian">Mohammad K. Azarian</a>, May 21 2008</div> <div class=sectline>a(n) is the number of permutations on [n+1] such that every initial segment is an interval of integers. Example: a(3) counts 1234, 2134, 2314, 2341, 3214, 3241, 3421, 4321. The map &quot;p -&gt; ascents of p&quot; is a bijection from these permutations to subsets of [n]. An ascent of a permutation p is a position i such that p(i) &lt; p(i+1). The permutations shown map to 123, 23, 13, 12, 3, 2, 1 and the empty set respectively. - <a href="/wiki/User:David_Callan">David Callan</a>, Jul 25 2008</div> <div class=sectline>2^(n-1) is the largest number having n divisors (in the sense of <a href="/A077569" title="Irregular triangle read by rows: row n lists numbers in the range 1 to 2^(n-1) (inclusive) that have exactly n divisors.">A077569</a>); <a href="/A005179" title="Smallest number with exactly n divisors.">A005179</a>(n) is the smallest. - <a href="/wiki/User:T._D._Noe">T. D. Noe</a>, Sep 02 2008</div> <div class=sectline>a(n) appears to match the number of divisors of the modified primorials (excluding 2, 3 and 5). Very limited range examined, PARI example shown. - <a href="/wiki/User:Bill_McEachen">Bill McEachen</a>, Oct 29 2008</div> <div class=sectline>Successive k such that phi(k)/k = 1/2, where phi is Euler's totient function. - <a href="/wiki/User:Artur_Jasinski">Artur Jasinski</a>, Nov 07 2008</div> <div class=sectline>A classical transform consists (for general a(n)) in swapping a(2n) and a(2n+1); examples for Jacobsthal <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a> and successive differences: <a href="/A092808" title="Pair reversal of Jacobsthal numbers.">A092808</a>, <a href="/A094359" title="Pair reversal of a Jacobsthal sequence.">A094359</a>, <a href="/A140505" title="Second differences of Jacobsthal sequence A001045, pairs with even and odd indices swapped.">A140505</a>. a(n) = <a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a> leads to 2, 1, 8, 4, 32, 16, ... = <a href="/A135520" title="a(n) = 4*a(n-2).">A135520</a>. - <a href="/wiki/User:Paul_Curtz">Paul Curtz</a>, Jan 05 2009</div> <div class=sectline>This is also the (L)-sieve transform of {2, 4, 6, 8, ..., 2n, ...} = <a href="/A005843" title="The nonnegative even numbers: a(n) = 2n.">A005843</a>. (See <a href="/A152009" title="(L)-sieve transform of {1,4,7,10,...,3n-2,...} (A016777)">A152009</a> for the definition of the (L)-sieve transform.) - <a href="/wiki/User:John_W._Layman">John W. Layman</a>, Jan 23 2009</div> <div class=sectline>a(n) = a(n-1)-th even natural number (<a href="/A005843" title="The nonnegative even numbers: a(n) = 2n.">A005843</a>) for n &gt; 1. - <a href="/wiki/User:Jaroslav_Krizek">Jaroslav Krizek</a>, Apr 25 2009</div> <div class=sectline>For n &gt;= 0, a(n) is the number of leaves in a complete binary tree of height n. For n &gt; 0, a(n) is the number of nodes in an n-cube. - <a href="/wiki/User:K.V.Iyer">K.V.Iyer</a>, May 04 2009</div> <div class=sectline>Permutations of n+1 elements where no element is more than one position right of its original place. For example, there are 4 such permutations of three elements: 123, 132, 213, and 312. The 8 such permutations of four elements are 1234, 1243, 1324, 1423, 2134, 2143, 3124, and 4123. - <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Jun 24 2009</div> <div class=sectline>Catalan transform of <a href="/A099087" title="Expansion of 1/(1 - 2*x + 2*x^2).">A099087</a>. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jun 29 2009</div> <div class=sectline>a(n) written in base 2: 1,10,100,1000,10000,..., i.e., (n+1) times 1, n times 0 (<a href="/A011557" title="Powers of 10: a(n) = 10^n.">A011557</a>(n)). - <a href="/wiki/User:Jaroslav_Krizek">Jaroslav Krizek</a>, Aug 02 2009</div> <div class=sectline>Or, phi(n) is equal to the number of perfect partitions of n. - <a href="/wiki/User:Juri-Stepan_Gerasimov">Juri-Stepan Gerasimov</a>, Oct 10 2009</div> <div class=sectline>These are the 2-smooth numbers, positive integers with no prime factors greater than 2. - <a href="/wiki/User:Michael_B._Porter">Michael B. Porter</a>, Oct 04 2009</div> <div class=sectline><a href="/A064614" title="Exchange 2 and 3 in the prime factorization of n.">A064614</a>(a(n)) = <a href="/A000244" title="Powers of 3: a(n) = 3^n.">A000244</a>(n) and <a href="/A064614" title="Exchange 2 and 3 in the prime factorization of n.">A064614</a>(m) &lt; <a href="/A000244" title="Powers of 3: a(n) = 3^n.">A000244</a>(n) for m &lt; a(n). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 08 2010</div> <div class=sectline>a(n) is the largest number m such that the number of steps of iterations of {r - (largest divisor d &lt; r)} needed to reach 1 starting at r = m is equal to n. Example (a(5) = 32): 32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4; 4 - 2 = 2; 2 - 1 = 1; number 32 has 5 steps and is the largest such number. See <a href="/A105017" title="Positions of records in A064097.">A105017</a>, <a href="/A064097" title="A quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n &gt; 1.">A064097</a>, <a href="/A175125" title="a(n) is the number of numbers m such that the number of iterations of r -&gt; r - (largest divisor d &lt; r) needed to reach 1 sta...">A175125</a>. - <a href="/wiki/User:Jaroslav_Krizek">Jaroslav Krizek</a>, Feb 15 2010</div> <div class=sectline>a(n) is the smallest proper multiple of a(n-1). - <a href="/wiki/User:Dominick_Cancilla">Dominick Cancilla</a>, Aug 09 2010</div> <div class=sectline>The powers-of-2 triangle T(n, k), n &gt;= 0 and 0 &lt;= k &lt;= n, begins with: {1}; {2, 4}; {8, 16, 32}; {64, 128, 256, 512}; ... . The first left hand diagonal T(n, 0) = <a href="/A006125" title="a(n) = 2^(n*(n-1)/2).">A006125</a>(n + 1), the first right hand diagonal T(n, n) = <a href="/A036442" title="a(n) = 2^((n-1)*(n+2)/2).">A036442</a>(n + 1) and the center diagonal T(2*n, n) = <a href="/A053765" title="a(n) = 4^(n^2 - n).">A053765</a>(n + 1). Some triangle sums, see <a href="/A180662" title="The Golden Triangle: T(n,k) = A001654(k) for n&gt;=0 and 0&lt;=k&lt;=n.">A180662</a>, are: Row1(n) = <a href="/A122743" title="Number of normalized polynomials of degree n in GF(2)[x,y].">A122743</a>(n), Row2(n) = <a href="/A181174" title="The &quot;Row2&quot; sums of the powers-of-2 triangle A000079">A181174</a>(n), Fi1(n) = <a href="/A181175" title="The &quot;Fi1&quot; sums of the powers-of-2 triangle A000079">A181175</a>(n), Fi2(2*n) = <a href="/A181175" title="The &quot;Fi1&quot; sums of the powers-of-2 triangle A000079">A181175</a>(2*n) and Fi2(2*n + 1) = 2*<a href="/A181175" title="The &quot;Fi1&quot; sums of the powers-of-2 triangle A000079">A181175</a>(2*n + 1). - <a href="/wiki/User:Johannes_W._Meijer">Johannes W. Meijer</a>, Oct 10 2010</div> <div class=sectline>Records in the number of prime factors. - <a href="/wiki/User:Juri-Stepan_Gerasimov">Juri-Stepan Gerasimov</a>, Mar 12 2011</div> <div class=sectline>Row sums of <a href="/A152538" title="Triangle read by rows, A027293 * (A152537 * 0^(n-k))">A152538</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 10 2008</div> <div class=sectline><a href="/A078719" title="Number of odd terms among n, f(n), f(f(n)), ...., 1 for the Collatz function (that is, until reaching &quot;1&quot; for the first time...">A078719</a>(a(n)) = 1; <a href="/A006667" title="Number of tripling steps to reach 1 from n in '3x+1' problem, or -1 if 1 is never reached.">A006667</a>(a(n)) = 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Oct 08 2011</div> <div class=sectline>The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n&gt;=1, a(n) equals the number of 2-colored compositions of n such that no adjacent parts have the same color. - <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, Nov 17 2011</div> <div class=sectline>Equals <a href="/A001405" title="a(n) = binomial(n, floor(n/2)).">A001405</a> convolved with its right-shifted variant: (1 + 2x + 4x^2 + ...) = (1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + ...) * (1 + x + x^2 + 2x^3 + 3x^4 + 6x^5 + ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Nov 23 2011</div> <div class=sectline>The number of odd-sized subsets of an n+1-set. For example, there are 2^3 odd-sized subsets of {1, 2, 3, 4}, namely {1}, {2}, {3}, {4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, and {2, 3, 4}. Also, note that 2^n = Sum_{k=1..floor((n+1)/2)} C(n+1, 2k-1). - <a href="/wiki/User:Dennis_P._Walsh">Dennis P. Walsh</a>, Dec 15 2011</div> <div class=sectline>a(n) is the number of 1's in any row of Pascal's triangle (mod 2) whose row number has exactly n 1's in its binary expansion (see <a href="/A007318" title="Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 &lt;= k &lt;= n.">A007318</a> and <a href="/A047999" title="Sierpi艅ski's [Sierpinski's] triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 2.">A047999</a>). (The result of putting together <a href="/A001316" title="Gould's sequence: a(n) = Sum_{k=0..n} (binomial(n,k) mod 2); number of odd entries in row n of Pascal's triangle (A007318); ...">A001316</a> and <a href="/A000120" title="1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).">A000120</a>.) - <a href="/wiki/User:Marcus_Jaiclin">Marcus Jaiclin</a>, Jan 31 2012</div> <div class=sectline><a href="/A204455" title="Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.">A204455</a>(k) = 1 if and only if k is in this sequence. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Feb 04 2012</div> <div class=sectline>For n&gt;=1 apparently the number of distinct finite languages over a unary alphabet, whose minimum regular expression has alphabetic width n (verified up to n=17), see the Gruber/Lee/Shallit link. - <a href="/wiki/User:Hermann_Gruber">Hermann Gruber</a>, May 09 2012</div> <div class=sectline>First differences of <a href="/A000225" title="a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)">A000225</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Feb 19 2013</div> <div class=sectline>This is the lexicographically earliest sequence which contains no arithmetic progression of length 3. - Daniel E. Frohardt, Apr 03 2013</div> <div class=sectline>a(n-2) is the number of bipartitions of {1..n} (i.e., set partitions into two parts) such that 1 and 2 are not in the same subset. - <a href="/wiki/User:Jon_Perry">Jon Perry</a>, May 19 2013</div> <div class=sectline>Numbers n such that the n-th cyclotomic polynomial has a root mod 2; numbers n such that the n-th cyclotomic polynomial has an even number of odd coefficients. - <a href="/wiki/User:Eric_M._Schmidt">Eric M. Schmidt</a>, Jul 31 2013</div> <div class=sectline>More is known now about non-power-of-2 &quot;Almost Perfect Numbers&quot; as described in Dagal. - <a href="/wiki/User:Jonathan_Vos_Post">Jonathan Vos Post</a>, Sep 01 2013</div> <div class=sectline>Number of symmetric Ferrers diagrams that fit into an n X n box. - <a href="/wiki/User:Graham_H._Hawkes">Graham H. Hawkes</a>, Oct 18 2013</div> <div class=sectline>Numbers n such that sigma(2n) = 2n + sigma(n). - <a href="/wiki/User:Jahangeer_Kholdi">Jahangeer Kholdi</a>, Nov 23 2013</div> <div class=sectline>a(1), ..., a(floor(n/2)) are all values of permanent on set of square (0,1)-matrices of order n&gt;=2 with row and column sums 2. - <a href="/wiki/User:Vladimir_Shevelev">Vladimir Shevelev</a>, Nov 26 2013</div> <div class=sectline>Numbers whose base-2 expansion has exactly one bit set to 1, and thus has base-2 sum of digits equal to one. - <a href="/wiki/User:Stanislav_Sykora">Stanislav Sykora</a>, Nov 29 2013</div> <div class=sectline><a href="/A072219" title="Any number n can be written uniquely in the form n = 2^k_1 - 2^k_2 + 2^k_3 - ... + 2^k_{2r+1} where the signs alternate, the...">A072219</a>(a(n)) = 1. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 20 2014</div> <div class=sectline>a(n) is the largest number k such that (k^n-2)/(k-2) is an integer (for n &gt; 1); (k^a(n)+1)/(k+1) is never an integer (for k &gt; 1 and n &gt; 0). - <a href="/wiki/User:Derek_Orr">Derek Orr</a>, May 22 2014</div> <div class=sectline>If x = <a href="/A083420" title="a(n) = 2*4^n - 1.">A083420</a>(n), y = a(n+1) and z = <a href="/A087289" title="a(n) = 2^(2*n+1) + 1.">A087289</a>(n), then x^2 + 2*y^2 = z^2. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jun 09 2014</div> <div class=sectline>The mini-sequence b(n) = least number k &gt; 0 such that 2^k ends in n identical digits is given by {1, 18, 39}. The repeating digits are {2, 4, 8} respectively. Note that these are consecutive powers of 2 (2^1, 2^2, 2^3), and these are the only powers of 2 (2^k, k &gt; 0) that are only one digit. Further, this sequence is finite. The number of n-digit endings for a power of 2 with n or more digits id 4*5^(n-1). Thus, for b(4) to exist, one only needs to check exponents up to 4*5^3 = 500. Since b(4) does not exist, it is clear that no other number will exist. - <a href="/wiki/User:Derek_Orr">Derek Orr</a>, Jun 14 2014</div> <div class=sectline>The least number k &gt; 0 such that 2^k ends in n consecutive decreasing digits is a 3-number sequence given by {1, 5, 25}. The consecutive decreasing digits are {2, 32, 432}. There are 100 different 3-digit endings for 2^k. There are no k-values such that 2^k ends in '987', '876', '765', '654', '543', '321', or '210'. The k-values for which 2^k ends in '432' are given by 25 mod 100. For k = 25 + 100*x, the digit immediately before the run of '432' is {4, 6, 8, 0, 2, 4, 6, 8, 0, 2, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus, we see the digit before '432' will never be a 5. So, this sequence is complete. - <a href="/wiki/User:Derek_Orr">Derek Orr</a>, Jul 03 2014</div> <div class=sectline>a(n) is the number of permutations of length n avoiding both 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at <a href="/A245898" title="Number of permutations avoiding 231 that can be realized on increasing unary-binary trees with n nodes.">A245898</a>. - <a href="/wiki/User:Manda_Riehl">Manda Riehl</a>, Aug 05 2014</div> <div class=sectline>Numbers n such that sigma(n) = sigma(2n) - phi(4n). - <a href="/wiki/User:Farideh_Firoozbakht">Farideh Firoozbakht</a>, Aug 14 2014</div> <div class=sectline>This is a B_2 sequence: for i &lt; j, differences a(j) - a(i) are all distinct. Here 2*a(n) &lt; a(n+1) + 1, so a(n) - a(0) &lt; a(n+1) - a(n). - <a href="/wiki/User:Thomas_Ordowski">Thomas Ordowski</a>, Sep 23 2014</div> <div class=sectline>a(n) counts n-walks (closed) on the graph G(1-vertex; 1-loop, 1-loop). - <a href="/wiki/User:David_Neil_McGrath">David Neil McGrath</a>, Dec 11 2014</div> <div class=sectline>a(n-1) counts walks (closed) on the graph G(1-vertex; 1-loop, 2-loop, 3-loop, 4-loop, ...). - <a href="/wiki/User:David_Neil_McGrath">David Neil McGrath</a>, Jan 01 2015</div> <div class=sectline>b(0) = 4; b(n+1) is the smallest number not in the sequence such that b(n+1) - Prod_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n) = a(n) for n &gt; 2. - <a href="/wiki/User:Derek_Orr">Derek Orr</a>, Jan 15 2015</div> <div class=sectline>a(n) counts the permutations of length n+2 whose first element is 2 such that the permutation has exactly one descent. - <a href="/wiki/User:Ran_Pan">Ran Pan</a>, Apr 17 2015</div> <div class=sectline>a(0)-a(30) appear, with a(26)-a(30) in error, in tablet M 08613 (see CDLI link) from the Old Babylonian period (c. 1900-1600 BC). - <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Sep 03 2015</div> <div class=sectline>Subsequence of <a href="/A028982" title="Squares and twice squares.">A028982</a> (the squares or twice squares sequence). - <a href="/wiki/User:Timothy_L._Tiffin">Timothy L. Tiffin</a>, Jul 18 2016</div> <div class=sectline><a href="/A000120" title="1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).">A000120</a>(a(n)) = 1. <a href="/A000265" title="Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.">A000265</a>(a(n)) = 1. <a href="/A000593" title="Sum of odd divisors of n.">A000593</a>(a(n)) = 1. - <a href="/wiki/User:Juri-Stepan_Gerasimov">Juri-Stepan Gerasimov</a>, Aug 16 2016</div> <div class=sectline>Number of monotone maps f : [0..n] -&gt; [0..n] which are order-increasing (i &lt;= f(i)) and idempotent (f(f(i)) = f(i)). In other words, monads on the n-th ordinal (seen as a posetal category). Any monad f determines a subset of [0..n] that contains n, by considering its set of monad algebras = fixed points { i | f(i) = i }. Conversely, any subset S of [0..n] containing n determines a monad on [0..n], by the function i |-&gt; min { j | i &lt;= j, j in S }. - <a href="/wiki/User:Noam_Zeilberger">Noam Zeilberger</a>, Dec 11 2016</div> <div class=sectline>Consider n points lying on a circle. Then for n&gt;=2 a(n-2) gives the number of ways to connect two adjacent points with nonintersecting chords. - <a href="/wiki/User:Anton_Zakharov">Anton Zakharov</a>, Dec 31 2016</div> <div class=sectline>Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Feb 07 2017</div> <div class=sectline>Also the number of independent vertex sets and vertex covers in the n-empty graph. - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Sep 21 2017</div> <div class=sectline>Also the number of maximum cliques in the n-halved cube graph for n &gt; 4. - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Dec 04 2017</div> <div class=sectline>Number of pairs of compositions of n corresponding to a seaweed algebra of index n-1. - <a href="/wiki/User:Nick_Mayers">Nick Mayers</a>, Jun 25 2018</div> <div class=sectline>The multiplicative group of integers modulo a(n) is cyclic if and only if n = 0, 1, 2. For n &gt;= 3, it is a product of two cyclic groups. - <a href="/wiki/User:Jianing_Song">Jianing Song</a>, Jun 27 2018</div> <div class=sectline>k^n is the determinant of n X n matrix M_(i, j) = binomial(k + i + j - 2, j) - binomial(i+j-2, j), in this case k=2. - <a href="/wiki/User:Tony_Foster_III">Tony Foster III</a>, May 12 2019</div> <div class=sectline>Solutions to the equation Phi(2n + 2*Phi(2n)) = 2n. - <a href="/wiki/User:M._Farrokhi_D._G.">M. Farrokhi D. G.</a>, Jan 03 2020</div> <div class=sectline>a(n-1) is the number of subsets of {1,2,...,n} which have an element that is the size of the set. For example, for n = 4, a(3) = 8 and the subsets are {1}, {1,2}, {2,3}, {2,4}, {1,2,3}, {1,3,4}, {2,3,4}, {1,2,3,4}. - <a href="/wiki/User:Enrique_Navarrete">Enrique Navarrete</a>, Nov 21 2020</div> <div class=sectline>a(n) is the number of self-inverse (n+1)-order permutations with 231-avoiding. E.g., a(3) = 8: [1234, 1243, 1324, 1432, 2134, 2143, 3214, 4321]. - <a href="/wiki/User:Yuchun_Ji">Yuchun Ji</a>, Feb 26 2021</div> <div class=sectline>For any fixed k &gt; 0, a(n) is the number of ways to tile a strip of length n+1 with tiles of length 1, 2, ... k, where the tile of length k can be black or white, with the restriction that the first tile cannot be black. - <a href="/wiki/User:Greg_Dresden">Greg Dresden</a> and Bora Bursal谋, Aug 31 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 1016.</div> <div class=sectline>Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.</div> <div class=sectline>Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton &amp; Co., NY &amp; London, 1997, 搂4.5 Logarithms and 搂8.1 Terminology, pp. 150, 264.</div> <div class=sectline>Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton University Press, Princeton, NJ. 1998, pp. 69-70.</div> <div class=sectline>J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>V. E. Tarakanov, Combinatorial problems on binary matrices, Combin. Analysis, MSU, 5 (1980), 4-15. (Russian)</div> <div class=sectline>S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>N. J. A. Sloane, <a href="/A000079/b000079.txt">Table of n, 2^n for n = 0..1000</a></div> <div class=sectline>Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].</div> <div class=sectline>Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/2003.11533">Wilf equivalences between vincular patterns in inversion sequences</a>, arXiv:2003.11533 [math.CO], 2020.</div> <div class=sectline>Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.</div> <div class=sectline>Jonathan Beagley and Lara Pudwell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Pudwell/pudwell13.html">Colorful Tilings and Permutations</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.</div> <div class=sectline>Arno Berger and Theodore P. Hill, <a href="http://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf">What is Benford's Law?</a>, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.</div> <div class=sectline>Tobias Boege and Thomas Kahle, <a href="https://arxiv.org/abs/1902.11260">Construction Methods for Gaussoids</a>, arXiv:1902.11260 [math.CO], 2019.</div> <div class=sectline>Anicius Manlius Severinus Boethius, <a href="http://www.e-codices.unifr.ch/en/vad/0296/6r/medium">De arithmetica</a>, Book 1, section 9.</div> <div class=sectline>Henry Bottomley, <a href="/A000079/a000079.gif">Illustration of initial terms</a></div> <div class=sectline>Douglas Butler, <a href="https://web.archive.org/web/20191202164323/http://www.tsm-resources.com:80/alists/pow2.html">Powers of Two up to 2^222</a></div> <div class=sectline>Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.</div> <div class=sectline>Peter J. Cameron, <a href="https://cameroncounts.wordpress.com/2017/05/15/notes-on-counting/">Notes on Counting</a>, Peter Cameron's Blog, 15/05/2017.</div> <div class=sectline>CDLI, <a href="http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&amp;ObjectID=P390441">M 08613</a>.</div> <div class=sectline>Giulio Cerbai, Anders Claesson, and Luca Ferrari, <a href="https://arxiv.org/abs/1907.08142">Stack sorting with restricted stacks</a>, arXiv:1907.08142 [cs.DS], 2019.</div> <div class=sectline>V. Coll, M. Hyatt, C. Magnant, and H. Wang, <a href="http://dx.doi.org/10.4172/1736-4337.1000227">Meander graphs and Frobenius seaweed Lie algebras II</a>, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.</div> <div class=sectline>M. Coons and H. Winning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Coons/coons2.html">Powers of Two Modulo Powers of Three</a>, J. Int. Seq. 18 (2015) # 15.6.1.</div> <div class=sectline>Keneth Adrian P. Dagal and Jose Arnaldo B. Dris, <a href="http://arxiv.org/abs/1308.6767">A Criterion for Almost Perfect Numbers in Terms of the Abundancy Index</a>, arXiv:1308.6767v1 [math.NT], Aug 14 2013.</div> <div class=sectline>V. Dergachev and A. Kirillov, <a href="https://www.emis.de/journals/JLT/vol.10_no.2/6.html">Index of Lie algebras of seaweed type</a>, J. Lie Theory 10 (2) (2000) 331-343.</div> <div class=sectline>Persi Diaconis, <a href="https://doi.org/10.1214/aop/1176995891">The distribution of leading digits and uniform distribution mod 1</a>, Ann. Probability, 5, 1977, 72--81.</div> <div class=sectline>Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.</div> <div class=sectline>David Eppstein, <a href="https://arxiv.org/abs/1804.07396">Making Change in 2048</a>, arXiv:1804.07396 [cs.DM], 2018.</div> <div class=sectline>P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 18</div> <div class=sectline>Jo毛l Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.</div> <div class=sectline>Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.</div> <div class=sectline>Hermann Gruber, Jonathan Lee and Jeffrey Shallit, <a href="http://arxiv.org/abs/1204.4982">Enumerating regular expressions and their languages</a>, arXiv:1204.4982v1 [cs.FL], 2012.</div> <div class=sectline>R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a></div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=6">Encyclopedia of Combinatorial Structures 6</a></div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=68">Encyclopedia of Combinatorial Structures 68</a></div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=72">Encyclopedia of Combinatorial Structures 72</a></div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=267">Encyclopedia of Combinatorial Structures 267</a></div> <div class=sectline>Marcus Jaiclin, et al. <a href="https://web.archive.org/web/20170823000349/http://pyrrho.wsc.ma.edu/math/faculty/jaiclin/writings/research/pascals_triangle/">Pascal's Triangle, Mod 2,3,5</a></div> <div class=sectline>Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a></div> <div class=sectline>J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.</div> <div class=sectline>P. A. MacMahon, <a href="http://www.jstor.org/stable/90632">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.</div> <div class=sectline>Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper &quot;Sorting numbers for cylinders...&quot;, letter to N. J. A. Sloane, N. D.</a></div> <div class=sectline>Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.</div> <div class=sectline>R. Ondrejka, <a href="http://www.jstor.org/stable/2004456">Exact values of 2^n, n=1(1)4000</a>, Math. Comp., 23 (1969), 456.</div> <div class=sectline>G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.</div> <div class=sectline>Robert Price, <a href="/A000079/a000079.txt">Comments on A000079 concerning Elementary Cellular Automata</a>, Feb 26 2016.</div> <div class=sectline>Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Saito/saito22.html">Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values</a>, J. Int. Seq. 14 (2011) # 11.2.4, Table 1.</div> <div class=sectline>Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.</div> <div class=sectline>J. Tanton, <a href="http://www.jstor.org/stable/25678324">A Dozen Questions about the Powers of Two</a>, Math Horizons, Vol. 8, pp. 5-10, September 2001.</div> <div class=sectline>G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Nombre/Puiss2.htm">Puissances de 2</a></div> <div class=sectline>Sage Weil, <a href="https://web.archive.org/web/20090206092940/http://newdream.net:80/~sage/old/numbers/pow2.htm">1058 powers of two</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundance.html">Abundance</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Hailstone Number (Collatz Problem)</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Composition.html">Composition</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EmptyGraph.html">Empty Graph</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erf.html">Erf</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FractionalPart.html">Fractional Part</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalvedCubeGraph.html">Halved Cube Graph</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hypercube.html">Hypercube</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeastDeficientNumber.html">Least Deficient Number</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerFractionalParts.html">PowerFractional Parts</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Subset.html">Subset</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a></div> <div class=sectline>Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_perfect_number">Almost perfect number</a></div> <div class=sectline>S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a></div> <div class=sectline><a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a></div> <div class=sectline><a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a></div> <div class=sectline><a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a></div> <div class=sectline><a href="/index/Cor#core">Index entries for &quot;core&quot; sequences</a></div> <div class=sectline><a href="/index/Di#divseq">Index to divisibility sequences</a></div> <div class=sectline><a href="/index/Par#partN">Index entries for related partition-counting sequences</a></div> <div class=sectline><a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).</div> <div class=sectline><a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = 2^n.</div> <div class=sectline>a(0) = 1; a(n) = 2*a(n-1).</div> <div class=sectline>G.f.: 1/(1 - 2*x).</div> <div class=sectline>E.g.f.: exp(2*x).</div> <div class=sectline>a(n)= Sum_{k = 0..n} binomial(n, k).</div> <div class=sectline>a(n) is the number of occurrences of n in <a href="/A000523" title="a(n) = floor(log_2(n)).">A000523</a>. a(n) = <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n) + <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n+1). a(n) = 1 + Sum_{k = 0..(n - 1)} a(k). The Hankel transform of this sequence gives <a href="/A000007" title="The characteristic function of {0}: a(n) = 0^n.">A000007</a> = [1, 0, 0, 0, 0, 0, ...]. - <a href="/wiki/User:Philippe_Del茅ham">Philippe Del茅ham</a>, Feb 25 2004</div> <div class=sectline>n such that phi(n) = n/2, for n &gt; 1, where phi is Euler's totient (<a href="/A000010" title="Euler totient function phi(n): count numbers &lt;= n and prime to n.">A000010</a>). - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, Sep 07 2004</div> <div class=sectline>a(n + 1) = a(n) XOR 3*a(n) where XOR is the binary exclusive OR operator. - <a href="/wiki/User:Philippe_Del茅ham">Philippe Del茅ham</a>, Jun 19 2005</div> <div class=sectline>a(n) = StirlingS2(n + 1, 2) + 1. - <a href="/wiki/User:Ross_La_Haye">Ross La Haye</a>, Jan 09 2008</div> <div class=sectline>a(n+2) = 6a(n+1) - 8a(n), n = 1, 2, 3, ... with a(1) = 1, a(2) = 2. - <a href="/wiki/User:Yosu_Yurramendi">Yosu Yurramendi</a>, Aug 06 2008</div> <div class=sectline>a(n) = ka(n-1) + (4 - 2k)a(n-2) for any integer k and n &gt; 1, with a(0) = 1, a(1) = 2. - <a href="/wiki/User:Jaume_Oliver_Lafont">Jaume Oliver Lafont</a>, Dec 05 2008</div> <div class=sectline>a(n) = Sum_{l_1 = 0..n + 1} Sum_{l_2 = 0..n}...Sum_{l_i = 0..n - i}...Sum_{l_n = 0..1} delta(l_1, l_2, ..., l_i, ..., l_n) where delta(l_1, l_2, ..., l_i, ..., l_n) = 0 if any l_i &lt;= l_(i+1) and l_(i+1) != 0 and delta(l_1, l_2, ..., l_i, ..., l_n) = 1 otherwise. - <a href="/wiki/User:Thomas_Wieder">Thomas Wieder</a>, Feb 25 2009</div> <div class=sectline>a(0) = 1, a(1) = 2; a(n) = a(n-1)^2/a(n-2), n &gt;= 2. - <a href="/wiki/User:Jaume_Oliver_Lafont">Jaume Oliver Lafont</a>, Sep 22 2009</div> <div class=sectline>a(n) = <a href="/A173786" title="Triangle read by rows: T(n,k) = 2^n + 2^k, 0 &lt;= k &lt;= n.">A173786</a>(n, n)/2 = <a href="/A173787" title="Triangle read by rows: T(n,k) = 2^n - 2^k, 0 &lt;= k &lt;= n.">A173787</a>(n + 1, n). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 28 2010</div> <div class=sectline>If p[i] = i - 1 and if A is the Hessenberg matrix of order n defined by: A[i, j] = p[j - i + 1], (i &lt;= j), A[i, j] = -1, (i = j + 1), and A[i, j] = 0 otherwise. Then, for n &gt;= 1, a(n-1) = det A. - <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, May 02 2010</div> <div class=sectline>If p[i] = Fibonacci(i-2) and if A is the Hessenberg matrix of order n defined by: A[i, j] = p[j - i + 1], (i &lt;= j), A[i, j] = -1, (i = j + 1), and A[i, j] = 0 otherwise. Then, for n &gt;= 2, a(n-2) = det A. - <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, May 08 2010</div> <div class=sectline>The sum of reciprocals, 1/1 + 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 2. - <a href="/wiki/User:Mohammad_K._Azarian">Mohammad K. Azarian</a>, Dec 29 2010</div> <div class=sectline>a(n) = 2*<a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n) + <a href="/A078008" title="Expansion of (1 - x)/((1 + x)*(1 - 2*x)).">A078008</a>(n) = 3*<a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n) + (-1)^n. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Feb 20 2003</div> <div class=sectline>a(n) = <a href="/A118654" title="Square array T(n,k) read by antidiagonals: T(n,k) = 2^n*Fibonacci(k) - Fibonacci(k-2).">A118654</a>(n, 2).</div> <div class=sectline>a(n) = <a href="/A140740" title="Triangle read by rows: T(n,n) = 1 and for k with 1 &lt;= k &lt; n: T(n+1,k) = T(n,k) + T(n - n mod k, k).">A140740</a>(n+1, 1).</div> <div class=sectline>a(n) = <a href="/A131577" title="Zero followed by powers of 2 (cf. A000079).">A131577</a>(n) + <a href="/A011782" title="Coefficients of expansion of (1-x)/(1-2*x) in powers of x.">A011782</a>(n) = <a href="/A024495" title="a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).">A024495</a>(n) + <a href="/A131708" title="A024494 prefixed by a 0.">A131708</a>(n) + <a href="/A024493" title="a(n) = C(n,0) + C(n,3) + ... + C(n,3[n/3]).">A024493</a>(n) = <a href="/A000749" title="a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n &gt; 3, with a(0)=a(1)=a(2)=0, a(3)=1.">A000749</a>(n) + <a href="/A038503" title="Sum of every 4th entry of row n in Pascal's triangle, starting at &quot;n choose 0&quot;.">A038503</a>(n) + <a href="/A038504" title="Sum of every 4th entry of row n in Pascal's triangle, starting at &quot;n choose 1&quot;.">A038504</a>(n) + <a href="/A038505" title="Sum of every 4th entry of row n in Pascal's triangle, starting at binomial(n,2).">A038505</a>(n) = <a href="/A139761" title="a(n) = Sum_{k &gt;= 0} binomial(n,5*k+4).">A139761</a>(n) + <a href="/A139748" title="a(n) = Sum_{ k &gt;= 0} binomial(n,5*k+3).">A139748</a>(n) + <a href="/A139714" title="a(n) = Sum_{k&gt;=0} binomial(n,5*k+2).">A139714</a>(n) + <a href="/A133476" title="a(n) = Sum_{k&gt;=0} binomial(n,5*k+1).">A133476</a>(n) + <a href="/A139398" title="a(n) = Sum_{k &gt;= 0} binomial(n,5*k).">A139398</a>(n). - <a href="/wiki/User:Paul_Curtz">Paul Curtz</a>, Jul 25 2011</div> <div class=sectline>a(n) = row sums of <a href="/A007318" title="Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 &lt;= k &lt;= n.">A007318</a>. - <a href="/wiki/User:Susanne_Wienand">Susanne Wienand</a>, Oct 21 2011</div> <div class=sectline>a(n) = Hypergeometric([-n], [], -1). - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Nov 01 2011</div> <div class=sectline>G.f.: A(x) = B(x)/x, B(x) satisfies B(B(x)) = x/(1 - x)^2. - <a href="/wiki/User:Vladimir_Kruchinin">Vladimir Kruchinin</a>, Nov 10 2011</div> <div class=sectline>a(n) = Sum_{k = 0..n} <a href="/A201730" title="Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is th...">A201730</a>(n, k)*(-1)^k. - <a href="/wiki/User:Philippe_Del茅ham">Philippe Del茅ham</a>, Dec 06 2011</div> <div class=sectline>2^n = Sum_{k = 1..floor((n+1)/2)} C(n+1, 2k-1). - <a href="/wiki/User:Dennis_P._Walsh">Dennis P. Walsh</a>, Dec 15 2011</div> <div class=sectline><a href="/A209229" title="Characteristic function of powers of 2, cf. A000079.">A209229</a>(a(n)) = 1. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Mar 07 2012</div> <div class=sectline><a href="/A001227" title="Number of odd divisors of n.">A001227</a>(a(n)) = 1. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, May 01 2012</div> <div class=sectline>Sum_{n &gt;= 1} mobius(n)/a(n) = 0.1020113348178103647430363939318... - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Aug 12 2012</div> <div class=sectline>E.g.f.: 1 + 2*x/(U(0) - x) where U(k) = 6*k + 1 + x^2/(6*k+3 + x^2/(6*k + 5 + x^2/U(k+1) )); (continued fraction, 3-step). - <a href="/wiki/User:Sergei_N._Gladkovskii">Sergei N. Gladkovskii</a>, Dec 04 2012</div> <div class=sectline>a(n) = det(|s(i+2,j)|, 1 &lt;= i,j &lt;= n), where s(n,k) are Stirling numbers of the first kind. - <a href="/wiki/User:Mircea_Merca">Mircea Merca</a>, Apr 04 2013</div> <div class=sectline>a(n) = det(|ps(i+1,j)|, 1 &lt;= i,j &lt;= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (<a href="/A129467" title="Orthogonal polynomials with all zeros integers from 2*A000217.">A129467</a>). - <a href="/wiki/User:Mircea_Merca">Mircea Merca</a>, Apr 06 2013</div> <div class=sectline>G.f.: W(0), where W(k) = 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/( 2*x*(k+2) + 1/W(k+1) )); (continued fraction). - <a href="/wiki/User:Sergei_N._Gladkovskii">Sergei N. Gladkovskii</a>, Aug 28 2013</div> <div class=sectline>a(n-1) = Sum_{t_1 + 2*t_2 + ... + n*t_n = n} multinomial(t_1 + t_2 + ... + t_n; t_1, t_2, ..., t_n). - <a href="/wiki/User:Mircea_Merca">Mircea Merca</a>, Dec 06 2013</div> <div class=sectline>Construct the power matrix T(n,j) = [A^*j]*[S^*(j-1)] where A(n)=(1,1,1,...) and S(n)=(0,1,0,0,...) (where * is convolution operation). Then a(n-1) = Sum_{j=1..n} T(n,j). - <a href="/wiki/User:David_Neil_McGrath">David Neil McGrath</a>, Jan 01 2015</div> <div class=sectline>a(n) = <a href="/A000005" title="d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.">A000005</a>(<a href="/A002110" title="Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.">A002110</a>(n)). - <a href="/wiki/User:Ivan_N._Ianakiev">Ivan N. Ianakiev</a>, May 23 2016</div> <div class=sectline>From <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Jul 18 2016: (Start)</div> <div class=sectline>Exponential convolution of <a href="/A000012" title="The simplest sequence of positive numbers: the all 1's sequence.">A000012</a> with themselves.</div> <div class=sectline>a(n) = Sum_{k=0..n} <a href="/A011782" title="Coefficients of expansion of (1-x)/(1-2*x) in powers of x.">A011782</a>(k).</div> <div class=sectline>Sum_{n&gt;=0} a(n)/n! = exp(2) = <a href="/A072334" title="Decimal expansion of e^2.">A072334</a>.</div> <div class=sectline>Sum_{n&gt;=0} (-1)^n*a(n)/n! = exp(-2) = <a href="/A092553" title="Decimal expansion of 1/e^2.">A092553</a>. (End)</div> <div class=sectline>G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = <a href="/A090129" title="Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.">A090129</a>(x) = (1 + 2x + 2x^2 + 4x^3 + 8x^4 + ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Sep 13 2016</div> <div class=sectline>a(n) = <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>(n + 1) + <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>(n) + Sum_{k = 0..n - 2} <a href="/A000045" title="Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.">A000045</a>(k + 1)*2^(n - 2 - k). - <a href="/wiki/User:Melvin_Peralta">Melvin Peralta</a>, Dec 22 2017</div> <div class=sectline>a(n) = 7*<a href="/A077020" title="a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.">A077020</a>(n)^2 + <a href="/A077021" title="a(n) is the unique odd positive solution y of 2^n = 7x^2 + y^2.">A077021</a>(n)^2, n&gt;=3. - <a href="/wiki/User:Ralf_Steiner">Ralf Steiner</a>, Aug 08 2021</div> <div class=sectline>a(n)= n + 1 + Sum_{k=3..n+1} (2*k-5)*J(n+2-k), where Jacobsthal number J(n) = <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n). - <a href="/wiki/User:Michael_A._Allen">Michael A. Allen</a>, Jan 12 2022</div> <div class=sectline>Integral_{x=0..Pi} cos(x)^n*cos(n*x) dx = Pi/a(n) (see Nahin, pp. 69-70). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, May 17 2023</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a> := n-&gt;2^n; [ seq(2^n, n=0..50) ];</div> <div class=sectline>isA000079 := proc(n)</div> <div class=sectline> local fs;</div> <div class=sectline> fs := numtheory[factorset](n) ;</div> <div class=sectline> if n = 1 then</div> <div class=sectline> true ;</div> <div class=sectline> elif nops(fs) &lt;&gt; 1 then</div> <div class=sectline> false;</div> <div class=sectline> elif op(1, fs) = 2 then</div> <div class=sectline> true;</div> <div class=sectline> else</div> <div class=sectline> false ;</div> <div class=sectline> end if;</div> <div class=sectline>end proc: # <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jan 09 2017</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[2^n, {n, 0, 50}]</div> <div class=sectline>2^Range[0, 50] (* <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 14 2014 *)</div> <div class=sectline>LinearRecurrence[{2}, {2}, {0, 20}] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Sep 21 2017 *)</div> <div class=sectline>CoefficientList[Series[1/(1 - 2 x), {x, 0, 20}], x] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Sep 21 2017 *)</div> <div class=sectline>NestList[2# &amp;, 1, 40] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Oct 07 2019 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) <a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a>(n)=2^n \\ Edited by <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Aug 27 2014</div> <div class=sectline>(PARI) unimodal(n)=local(x, d, um, umc); umc=0; for (c=0, n!-1, x=numtoperm(n, c); d=0; um=1; for (j=2, n, if (x[j]&lt;x[j-1], d=1); if (x[j]&gt;x[j-1] &amp;&amp; d==1, um=0); if (um==0, break)); if (um==1, print(x)); umc+=um); umc</div> <div class=sectline>(Haskell)</div> <div class=sectline>a000079 = (2 ^)</div> <div class=sectline>a000079_list = iterate (* 2) 1</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 22 2014, Mar 05 2012, Dec 29 2011</div> <div class=sectline>(Maxima) <a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a>(n):=2^n$ makelist(<a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a>(n), n, 0, 30); /* <a href="/wiki/User:Martin_Ettl">Martin Ettl</a>, Nov 05 2012 */</div> <div class=sectline>(Magma) [2^n: n in [0..40]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Feb 17 2014</div> <div class=sectline>(Magma) [n le 2 select n else 5*Self(n-1)-6*Self(n-2): n in [1..40]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Feb 17 2014</div> <div class=sectline>(Scheme) (define (<a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a> n) (expt 2 n)) ;; <a href="/wiki/User:Antti_Karttunen">Antti Karttunen</a>, Mar 21 2017</div> <div class=sectline>(Scala) (List.fill(20)(2: BigInt)).scanLeft(1: BigInt)(_ * _) // <a href="/wiki/User:Alonso_del_Arte">Alonso del Arte</a>, Jan 16 2020</div> <div class=sectline>(Python)</div> <div class=sectline>def a(n): return 1&lt;&lt;n</div> <div class=sectline>print([a(n) for n in range(34)]) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Jul 28 2022</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000225" title="a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)">A000225</a>, <a href="/A038754" title="a(2n) = 3^n, a(2n+1) = 2*3^n.">A038754</a>, <a href="/A133464" title="a(3n)=4^n, a(3n+1)=2*4^n, a(3n+2)=3*4^n.">A133464</a>, <a href="/A140730" title="a(4*n)=5^n, a(4*n+1)=2*5^n, a(4*n+2)=3*5^n, a(4*n+3)=4*5^n.">A140730</a>, <a href="/A037124" title="Numbers that contain only one nonzero digit.">A037124</a>, <a href="/A001787" title="a(n) = n*2^(n-1).">A001787</a>, <a href="/A001788" title="a(n) = n*(n+1)*2^(n-2).">A001788</a>, <a href="/A001789" title="a(n) = binomial(n,3)*2^(n-3).">A001789</a>, <a href="/A003472" title="a(n) = 2^(n-4)*C(n,4).">A003472</a>, <a href="/A054849" title="a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.">A054849</a>, <a href="/A002409" title="a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.">A002409</a>, <a href="/A054851" title="a(n) = 2^(n-7)*binomial(n,7). Number of 7D hypercubes in an n-dimensional hypercube.">A054851</a>, <a href="/A140325" title="a(n) = binomial(n+8,8) * 2^n.">A140325</a>, <a href="/A140354" title="a(n) = binomial(n+9,9)*2^n.">A140354</a>, <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>, <a href="/A152537" title="Convolution sequence: this sequence convolved with A000041 gives powers of 2, (A000079).">A152537</a>, <a href="/A001405" title="a(n) = binomial(n, floor(n/2)).">A001405</a>, <a href="/A007318" title="Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 &lt;= k &lt;= n.">A007318</a>, <a href="/A000120" title="1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).">A000120</a>, <a href="/A000265" title="Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.">A000265</a>, <a href="/A000593" title="Sum of odd divisors of n.">A000593</a>, <a href="/A001227" title="Number of odd divisors of n.">A001227</a>, <a href="/A077020" title="a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.">A077020</a>, <a href="/A077021" title="a(n) is the unique odd positive solution y of 2^n = 7x^2 + y^2.">A077021</a>.</div> <div class=sectline>This is the Hankel transform (see <a href="/A001906" title="F(2n) = bisection of Fibonacci sequence: a(n) = 3*a(n-1) - a(n-2).">A001906</a> for the definition) of <a href="/A000984" title="Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.">A000984</a>, <a href="/A002426" title="Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.">A002426</a>, <a href="/A026375" title="a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).">A026375</a>, <a href="/A026387" title="a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).">A026387</a>, <a href="/A026569" title="a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.">A026569</a>, <a href="/A026585" title="a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.">A026585</a>, <a href="/A026671" title="Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).">A026671</a> and <a href="/A032351" title="Number of permutations of length n which avoid the patterns 2143, 1324 (smooth permutations); or avoid the patterns 1342, 24...">A032351</a>. - <a href="/wiki/User:John_W._Layman">John W. Layman</a>, Jul 31 2000</div> <div class=sectline>Euler transform of <a href="/A001037" title="Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over i...">A001037</a>, <a href="/A209406" title="Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.">A209406</a> (multisets), inverse binomial transform of <a href="/A000244" title="Powers of 3: a(n) = 3^n.">A000244</a>, binomial transform of <a href="/A000012" title="The simplest sequence of positive numbers: the all 1's sequence.">A000012</a>.</div> <div class=sectline>Complement of <a href="/A057716" title="The nonpowers of 2.">A057716</a>.</div> <div class=sectline>Boustrophedon transforms: <a href="/A000734" title="Boustrophedon transform of 1,1,2,4,8,16,32,...">A000734</a>, <a href="/A000752" title="Boustrophedon transform of powers of 2.">A000752</a>.</div> <div class=sectline>Range of values of <a href="/A006519" title="Highest power of 2 dividing n.">A006519</a>, <a href="/A007875" title="Number of ways of writing n as p*q, with p &lt;= q, gcd(p, q) = 1.">A007875</a>, <a href="/A011782" title="Coefficients of expansion of (1-x)/(1-2*x) in powers of x.">A011782</a>, <a href="/A030001" title="Smallest power of 2 whose decimal expansion contains n.">A030001</a>, <a href="/A034444" title="a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).">A034444</a>, <a href="/A037445" title="Number of infinitary divisors (or i-divisors) of n.">A037445</a>, <a href="/A053644" title="Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the firs...">A053644</a>, and <a href="/A054243" title="Number of partitions of n into distinct positive parts &lt;= n, where parts are combined by XOR.">A054243</a>.</div> <div class=sectline>Cf. <a href="/A018900" title="Sums of two distinct powers of 2.">A018900</a>, <a href="/A014311" title="Numbers with exactly 3 ones in binary expansion.">A014311</a>, <a href="/A014312" title="Numbers with exactly 4 ones in binary expansion.">A014312</a>, <a href="/A014313" title="Numbers with exactly 5 ones in binary expansion.">A014313</a>, <a href="/A023688" title="Numbers with exactly 6 ones in binary expansion.">A023688</a>, <a href="/A023689" title="Numbers with exactly 7 ones in binary expansion.">A023689</a>, <a href="/A023690" title="Numbers with exactly 8 ones in binary expansion.">A023690</a>, <a href="/A023691" title="Numbers with exactly 9 ones in binary expansion.">A023691</a> (sum of 2, ..., 9 distinct powers of 2).</div> <div class=sectline>Cf. <a href="/A090129" title="Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.">A090129</a>.</div> <div class=sectline>The following are parallel families: <a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a> (2^n), <a href="/A004094" title="Powers of 2 written backwards.">A004094</a> (2^n reversed), <a href="/A028909" title="Arrange digits of 2^n in ascending order.">A028909</a> (2^n sorted up), <a href="/A028910" title="Arrange digits of 2^n in descending order.">A028910</a> (2^n sorted down), <a href="/A036447" title="Double and reverse digits.">A036447</a> (double and reverse), <a href="/A057615" title="ATS: Add Then Sort (i.e., double previous term and then sort digits).">A057615</a> (double and sort up), <a href="/A263451" title="a(n) is the largest anagram of 2*a(n-1), a(1)=1.">A263451</a> (double and sort down); <a href="/A000244" title="Powers of 3: a(n) = 3^n.">A000244</a> (3^n), <a href="/A004167" title="Powers of 3 written backwards.">A004167</a> (3^n reversed), <a href="/A321540" title="3^n with digits rearranged into nondecreasing order.">A321540</a> (3^n sorted up), <a href="/A321539" title="3^n with digits rearranged into nonincreasing order.">A321539</a> (3^n sorted down), <a href="/A163632" title="Triple and reverse digits.">A163632</a> (triple and reverse), <a href="/A321542" title="a(0)=1; thereafter a(n) = 3*a(n-1) with digits rearranged into nondecreasing order.">A321542</a> (triple and sort up), <a href="/A321541" title="a(0)=1; thereafter a(n) = 3*a(n-1) with digits rearranged into nonincreasing order.">A321541</a> (triple and sort down).</div> <div class=sectline>Sequence in context: <a href="/A123344" title="Expansion of (1+3*x)/(1+2*x).">A123344</a> <a href="/A141531" title="Inverse binomial transform of A001651.">A141531</a> <a href="/A084633" title="Inverse binomial transform of repeated odd numbers.">A084633</a> * <a href="/A011782" title="Coefficients of expansion of (1-x)/(1-2*x) in powers of x.">A011782</a> <a href="/A120617" title="Hankel transform of g.f. 1/sqrt(1+4x^2).">A120617</a> <a href="/A131577" title="Zero followed by powers of 2 (cf. A000079).">A131577</a></div> <div class=sectline>Adjacent sequences: <a href="/A000076" title="Number of integers &lt;= 2^n of form 4 x^2 + 4 x y + 5 y^2.">A000076</a> <a href="/A000077" title="Number of positive integers &lt;= 2^n of form x^2 + 6 y^2.">A000077</a> <a href="/A000078" title="Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) for n &gt;= 4 with a(0) = a(1) = a(2) = 0 and a(3) = 1.">A000078</a> * <a href="/A000080" title="Number of nonisomorphic minimal triangle graphs.">A000080</a> <a href="/A000081" title="Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).">A000081</a> <a href="/A000082" title="a(n) = n^2*Product_{p|n} (1 + 1/p).">A000082</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="an important sequence">core</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Clarified a comment <a href="/wiki/User:T._D._Noe">T. D. Noe</a>, Aug 30 2009</div> <div class=sectline>Edited by <a href="/wiki/User:Daniel_Forgues">Daniel Forgues</a>, May 12 2010</div> <div class=sectline>Incorrect comment deleted by <a href="/wiki/User:Matthew_Vandermast">Matthew Vandermast</a>, May 17 2014</div> <div class=sectline>Comment corrected to match offset by <a href="/wiki/User:Geoffrey_Critzer">Geoffrey Critzer</a>, Nov 28 2014</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 17 16:23 EST 2025. Contains 380975 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>

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