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A001105 - 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>A001105 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A001105" 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=%2fA001105">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="A001105 - 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> A001105 </div> <div class=seqname> a(n) = 2*n^2. </div> </div> <div class=scorerefs> 230 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882, 968, 1058, 1152, 1250, 1352, 1458, 1568, 1682, 1800, 1922, 2048, 2178, 2312, 2450, 2592, 2738, 2888, 3042, 3200, 3362, 3528, 3698, 3872, 4050, 4232, 4418</div> <div class=seqdatalinks> (<a href="/A001105/list">list</a>; <a href="/A001105/graph">graph</a>; <a href="/search?q=A001105+-id:A001105">refs</a>; <a href="/A001105/listen">listen</a>; <a href="/history?seq=A001105">history</a>; <a href="/search?q=id:A001105&fmt=text">text</a>; <a href="/A001105/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>Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - <a href="/wiki/User:Roberto_E._Martinez_II">Roberto E. Martinez II</a>, Jan 07 2002</div> <div class=sectline>"If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature, Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.</div> <div class=sectline>Let z(1) = i = sqrt(-1), z(k+1) = 1/(z(k)+2i); then a(n) = (-1)*Imag(z(n+1))/Real(z(n+1)). - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Aug 06 2002</div> <div class=sectline>Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of <a href="/A016825" title="Positive integers congruent to 2 (mod 4): a(n) = 4*n+2, for n >= 0.">A016825</a>. - <a href="/wiki/User:Jeremy_Gardiner">Jeremy Gardiner</a>, Dec 19 2004</div> <div class=sectline>Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2, (15+21)/2, ... . - <a href="/wiki/User:Amarnath_Murthy">Amarnath Murthy</a>, Aug 05 2005</div> <div class=sectline>These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010</div> <div class=sectline>Integral areas of isosceles right triangles with rational legs (legs are 2n and triangles are nondegenerate for n > 0). - <a href="/wiki/User:Rick_L._Shepherd">Rick L. Shepherd</a>, Sep 29 2009</div> <div class=sectline>Even squares divided by 2. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Aug 18 2011</div> <div class=sectline>Number of stars when distributed as in the U.S.A. flag: n rows with n+1 stars and, between each pair of these, one row with n stars (i.e., n-1 of these), i.e., n*(n+1)+(n-1)*n = 2*n^2 = <a href="/A001105" title="a(n) = 2*n^2.">A001105</a>(n). - <a href="/wiki/User:C茅sar_Eliud_Lozada">C茅sar Eliud Lozada</a>, Sep 17 2012</div> <div class=sectline>Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak having height n-3. - <a href="/wiki/User:David_Scambler">David Scambler</a>, Apr 29 2013</div> <div class=sectline>Sum of the partition parts of 2n into exactly two parts. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 01 2013</div> <div class=sectline>Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (<a href="/A020882" title="Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles.">A020882</a>) and respective odd leg a (<a href="/A180620" title="Odd legs of primitive Pythagorean triples (with multiplicity) sorted with respect to increasing hypotenuse.">A180620</a>); sequence gives values c-a, sorted with duplicates removed. - <a href="/wiki/User:K._G._Stier">K. G. Stier</a>, Nov 04 2013</div> <div class=sectline>Number of roots in the root systems of type B_n and C_n (for n > 1). - <a href="/wiki/User:Tom_Edgar">Tom Edgar</a>, Nov 05 2013</div> <div class=sectline>Area of a square with diagonal 2n. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 18 2014</div> <div class=sectline>This sequence appears also as the first and second member of the quartet [a(n), a(n), p(n), p(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p(n) = <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n). See an Oct 15 2014 comment on <a href="/A147973" title="a(n) = -2*n^2 + 12*n - 14.">A147973</a> where also a reference is given. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 16 2014</div> <div class=sectline>a(n) are the only integers m where (<a href="/A000005" title="d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.">A000005</a>(m) + <a href="/A000203" title="a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).">A000203</a>(m)) = (number of divisors of m + sum of divisors of m) is an odd number. - <a href="/wiki/User:Richard_R._Forberg">Richard R. Forberg</a>, Jan 09 2015</div> <div class=sectline>a(n) represents the first term in a sum of consecutive integers running to a(n+1)-1 that equals (2n+1)^3. - <a href="/wiki/User:Patrick_J._McNab">Patrick J. McNab</a>, Dec 24 2016</div> <div class=sectline>Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Jul 29 2017</div> <div class=sectline>Also the Wiener index of the n-cocktail party graph for n > 1. - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Sep 07 2017</div> <div class=sectline>Numbers represented as the palindrome 242 in number base B including B=2 (binary), 3 (ternary) and 4: 242(2)=18, 242(3)=32, 242(4)=50, ... 242(9)=200, 242(10)=242, ... - <a href="/wiki/User:Ron_Knott">Ron Knott</a>, Nov 14 2017</div> <div class=sectline>a(n) is the square of the hypotenuse of an isosceles right triangle whose sides are equal to n. - <a href="/wiki/User:Thomas_M._Green">Thomas M. Green</a>, Aug 20 2019</div> <div class=sectline>The sequence contains all odd powers of 2 (<a href="/A004171" title="a(n) = 2^(2n+1).">A004171</a>) but no even power of 2 (<a href="/A000302" title="Powers of 4: a(n) = 4^n.">A000302</a>). - <a href="/wiki/User:Torlach_Rush">Torlach Rush</a>, Oct 10 2019</div> <div class=sectline>From <a href="/wiki/User:Bernard_Schott">Bernard Schott</a>, Aug 31 2021 and Sep 16 2021: (Start)</div> <div class=sectline>Apart from 0, integers such that the number of even divisors (<a href="/A183063" title="Number of even divisors of n.">A183063</a>) is odd.</div> <div class=sectline>Proof: every n = 2^q * (2k+1), q, k >= 0, then 2*n^2 = 2^(2q+1) * (2k+1)^2; now, gcd(2, 2k+1) = 1, tau(2^(2q+1)) = 2q+2 and tau((2k+1)^2) = 2u+1 because (2k+1)^2 is square, so, tau(2*n^2) = (2q+2) * (2u+1).</div> <div class=sectline>The 2q+2 divisors of 2^(2q+1) are {1, 2, 2^2, 2^3, ..., 2^(2q+1)}, so 2^(2q+1) has 2q+1 even divisors {2^1, 2^2, 2^3, ..., 2^(2q+1)}.</div> <div class=sectline>Conclusion: these 2q+1 even divisors create with the 2u+1 odd divisors of (2k+1)^2 exactly (2q+1)*(2u+1) even divisors of 2*n^2, and (2q+1)*(2u+1) is odd. (End)</div> <div class=sectline>a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - <a href="/wiki/User:Paul_Weisenhorn">Paul Weisenhorn</a>, Jan 29 2022</div> <div class=sectline>Number of points at L1 distance = 2 from any given point in Z^n. - <a href="/wiki/User:Shel_Kaphan">Shel Kaphan</a>, Feb 25 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.</div> <div class=sectline>Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles, Paradoxes and Problems, Chapter 2 entitled "The Calculus of Finite Differences," W. W. Norton and Company, New York, 2001, pages 12-13.</div> <div class=sectline>L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.</div> <div class=sectline>Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000, p. 213.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A001105/b001105.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.</div> <div class=sectline>Milan Janji膰, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8.</div> <div class=sectline>Milan Janji膰, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.</div> <div class=sectline>Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Feb 13 2013</div> <div class=sectline>Vladimir Ladma, <a href="http://www.traced-ideas.eu/atom/atomcore.html">Magic Numbers</a>.</div> <div class=sectline>Vladimir Pletser, <a href="http://arxiv.org/abs/1501.06098">General solutions of sums of consecutive cubed integers equal to squared integers</a>, arXiv:1501.06098 [math.NT], 2015.</div> <div class=sectline>Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv:1406.3081 [math.CO], 2014.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.</div> <div class=sectline><a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.</div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (-1)^(n+1) * <a href="/A053120" title="Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).">A053120</a>(2*n, 2).</div> <div class=sectline>G.f.: 2*x*(1+x)/(1-x)^3.</div> <div class=sectline>a(n) = <a href="/A100345" title="Triangle read by rows: T(n,k) = n*(n+k), 0<=k<=n.">A100345</a>(n, n).</div> <div class=sectline>Sum_{n>=1} 1/a(n) = Pi^2/12 =<a href="/A013661" title="Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.">A013661</a>/2. [Jolley eq. 319]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 21 2006</div> <div class=sectline>a(n) = <a href="/A049452" title="Pentagonal numbers with even index.">A049452</a>(n) - <a href="/A033991" title="a(n) = n*(4*n-1).">A033991</a>(n). - <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Jun 12 2007</div> <div class=sectline>a(n) = <a href="/A016742" title="Even squares: a(n) = (2*n)^2.">A016742</a>(n)/2. - <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Jun 20 2008</div> <div class=sectline>a(n) = 2 * <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>(n). - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, May 14 2008</div> <div class=sectline>a(n) = 4*n + a(n-1) - 2, n > 0. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a></div> <div class=sectline>a(n) = <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n-1) + <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n). - Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010 [Corrected by <a href="/wiki/User:Klaus_Purath">Klaus Purath</a>, Jun 18 2020]</div> <div class=sectline>a(n) = <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>(n,k) + <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>(n,n-k+1), 1 <= k <= n. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 13 2010</div> <div class=sectline>a(n) = <a href="/A007607" title="Skip 1, take 2, skip 3, etc.">A007607</a>(<a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>(n)). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 12 2011</div> <div class=sectline>For n > 0, a(n) = 1/coefficient of x^2 in the Maclaurin expansion of 1/(cos(x)+n-1). - <a href="/wiki/User:Francesco_Daddi">Francesco Daddi</a>, Aug 04 2011</div> <div class=sectline>a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - <a href="/wiki/User:Artur_Jasinski">Artur Jasinski</a>, Nov 24 2011</div> <div class=sectline>a(n) = <a href="/A070216" title="Triangle T(n, k) = n^2 + k^2, 1 <= k <= n, read by rows.">A070216</a>(n,n) for n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 11 2012</div> <div class=sectline>a(n) = <a href="/A014132" title="Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + n-k)/2, n, k > 0, read by antidiagonals.">A014132</a>(2*n-1,n) for n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 12 2012</div> <div class=sectline>a(n) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) + <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a>(n). - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 11 2013</div> <div class=sectline>(a(n) - <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(k))^2 = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(2*n-1-k)*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(2*n+k) + n^2, for all k. - <a href="/wiki/User:Charlie_Marion">Charlie Marion</a>, May 04 2013</div> <div class=sectline>a(n) = floor(1/(1-cos(1/n))), n > 0. - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Oct 08 2014</div> <div class=sectline>a(n) = <a href="/A251599" title="Centers of rows of the triangular array formed by the natural numbers.">A251599</a>(3*n-1) for n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 13 2014</div> <div class=sectline>a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Mar 12 2015</div> <div class=sectline>a(n) = <a href="/A002061" title="Central polygonal numbers: a(n) = n^2 - n + 1.">A002061</a>(n+1) + <a href="/A165900" title="a(n) = n^2 - n - 1.">A165900</a>(n). - <a href="/wiki/User:Torlach_Rush">Torlach Rush</a>, Feb 21 2019</div> <div class=sectline>E.g.f.: 2*exp(x)*x*(1 + x). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Oct 12 2019</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (<a href="/A222171" title="Decimal expansion of Pi^2/24.">A222171</a>). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jul 03 2020</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 03 2021: (Start)</div> <div class=sectline>Product_{n>=1} (1 + 1/a(n)) = sqrt(2)*sinh(Pi/sqrt(2))/Pi.</div> <div class=sectline>Product_{n>=1} (1 - 1/a(n)) = sqrt(2)*sin(Pi/sqrt(2))/Pi. (End)</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(3) = 18; since 2(3) = 6 has 3 partitions with exactly two parts: (5,1), (4,2), (3,3). Adding all the parts, we get: 1 + 2 + 3 + 3 + 4 + 5 = 18. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 01 2013</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A001105" title="a(n) = 2*n^2.">A001105</a>:=n->2*n^2; seq(<a href="/A001105" title="a(n) = 2*n^2.">A001105</a>(k), k=0..100); # <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Oct 29 2013</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>2 Range[0, 50]^2 (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jan 23 2011 *)</div> <div class=sectline>LinearRecurrence[{3, -3, 1}, {2, 8, 18}, {0, 20}] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Jul 28 2017 *)</div> <div class=sectline>2 PolygonalNumber[4, Range[0, 20]] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Jul 28 2017 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) [2*n^2: n in [0..50] ]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Apr 30 2011</div> <div class=sectline>(PARI) a(n) = 2*n^2; \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jun 16 2011</div> <div class=sectline>(Haskell)</div> <div class=sectline>a001105 = a005843 . a000290 -- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 12 2012</div> <div class=sectline>(Sage) [2*n^2 for n in (0..20)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Feb 22 2019</div> <div class=sectline>(GAP) List([0..50], n->2*n^2); # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Feb 24 2019</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A006331" title="a(n) = n*(n+1)*(2*n+1)/3.">A006331</a> (partial sums), <a href="/A016742" title="Even squares: a(n) = (2*n)^2.">A016742</a>, <a href="/A056106" title="Second spoke of a hexagonal spiral.">A056106</a>, <a href="/A116471" title="Values 2*(n -+ 1)^2 sorted.">A116471</a>, <a href="/A245508" title="Smallest double square (cf. A001105) greater than n-th prime.">A245508</a>, <a href="/A251599" title="Centers of rows of the triangular array formed by the natural numbers.">A251599</a>, <a href="/A002061" title="Central polygonal numbers: a(n) = n^2 - n + 1.">A002061</a>, <a href="/A165900" title="a(n) = n^2 - n - 1.">A165900</a>.</div> <div class=sectline>Cf. numbers of the form n*(n*k-k+4)/2 listed in <a href="/A226488" title="a(n) = n*(13*n - 9)/2.">A226488</a>.</div> <div class=sectline>Cf. <a href="/A058331" title="a(n) = 2*n^2 + 1.">A058331</a> and <a href="/A247375" title="Numbers m such that floor(m/2) is a square.">A247375</a>. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Sep 16 2014</div> <div class=sectline>Cf. <a href="/A194715" title="15 times triangular numbers.">A194715</a> (4-cycles in the triangular honeycomb obtuse knight graph), <a href="/A290391" title="Number of 5-cycles in the n-triangular honeycomb obtuse knight graph.">A290391</a> (5-cycles), <a href="/A290392" title="Number of 6-cycles in the n-triangular honeycomb obtuse knight graph.">A290392</a> (6-cycles). - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Jul 29 2017</div> <div class=sectline>Cf. <a href="/A139098" title="a(n) = 8*n^2.">A139098</a>, <a href="/A077591" title="Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals.">A077591</a>.</div> <div class=sectline>Cf. <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>, <a href="/A002266" title="Integers repeated 5 times.">A002266</a>.</div> <div class=sectline>Integers such that: this sequence (the number of even divisors is odd), <a href="/A028982" title="Squares and twice squares.">A028982</a> (the number of odd divisors is odd), <a href="/A028983" title="Numbers whose sum of divisors is even.">A028983</a> (the number of odd divisors is even), <a href="/A183300" title="Positive integers not of the form 2n^2.">A183300</a> (the number of even divisors is even).</div> <div class=sectline>Sequence in context: <a href="/A067051" title="The smallest k>1 such that k divides sigma(k*n) is equal to 3.">A067051</a> <a href="/A074629" title="Duplicate of A067051">A074629</a> <a href="/A209303" title="Numbers of the form x^2 + SumOfSquaredDigits(x).">A209303</a> * <a href="/A361905" title="Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.">A361905</a> <a href="/A336489" title="Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(...">A336489</a> <a href="/A051787" title="Expected number of spins in a 2-player Dreidel game (rounded to nearest integer) (version 1).">A051787</a></div> <div class=sectline>Adjacent sequences: <a href="/A001102" title="Numbers k such that k / (sum of digits of k) is a square.">A001102</a> <a href="/A001103" title="Numbers k such that (k / product of digits of k) is 1 or a prime.">A001103</a> <a href="/A001104" title="Numbers n such that n / product of digits of n is a square.">A001104</a> * <a href="/A001106" title="9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.">A001106</a> <a href="/A001107" title="10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).">A001107</a> <a href="/A001108" title="a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1.">A001108</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="it is very easy to produce terms of sequence">easy</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline>Bernd.Walter(AT)frankfurt.netsurf.de</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 November 28 12:26 EST 2024. Contains 378201 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>