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A016754 - 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>A016754 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A016754" 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=%2fA016754">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="A016754 - 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> A016754 </div> <div class=seqname> Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers. </div> </div> <div class=scorerefs> 296 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025</div> <div class=seqdatalinks> (<a href="/A016754/list">list</a>; <a href="/A016754/graph">graph</a>; <a href="/search?q=A016754+-id:A016754">refs</a>; <a href="/A016754/listen">listen</a>; <a href="/history?seq=A016754">history</a>; <a href="/search?q=id:A016754&fmt=text">text</a>; <a href="/A016754/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>The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year. - <a href="/wiki/User:Hans_Isdahl">Hans Isdahl</a>, Jan 26 2008</div> <div class=sectline>Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (<a href="/A000594" title="Ramanujan's tau function (or Ramanujan numbers, or tau numbers).">A000594</a>). - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, May 01 2003</div> <div class=sectline>If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, Oct 21 2007</div> <div class=sectline>Binomial transform of [1, 8, 8, 0, 0, 0, ...]; Narayana transform (<a href="/A001263" title="Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.">A001263</a>) of [1, 8, 0, 0, 0, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 29 2007</div> <div class=sectline>All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares see <a href="/A138393" title="Numbers of form 8k+1 which are not squares.">A138393</a>. Numbers 8k+1 are squares iff k is a triangular number from <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>. And squares have form 4n(n+1)+1. - <a href="/wiki/User:Artur_Jasinski">Artur Jasinski</a>, Mar 27 2008</div> <div class=sectline>Sequence arises from reading the line from 1, in the direction 1, 25, ... and the line from 9, in the direction 9, 49, ..., in the square spiral whose vertices are the squares <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, May 24 2008</div> <div class=sectline>Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a> & <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, May 29 2009</div> <div class=sectline>First differences: <a href="/A008590" title="Multiples of 8.">A008590</a>(n) = a(n) - a(n-1) for n>0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 08 2009</div> <div class=sectline>Central terms of the triangle in <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>; cf. <a href="/A000466" title="a(n) = 4*n^2 - 1.">A000466</a>, <a href="/A053755" title="a(n) = 4*n^2 + 1.">A053755</a>. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 13 2010</div> <div class=sectline>Odd numbers with odd abundance. Odd numbers with even abundance are in <a href="/A088828" title="Nonsquare positive odd numbers.">A088828</a>. Even numbers with odd abundance are in <a href="/A088827" title="Even numbers with odd abundance: even squares or two times squares.">A088827</a>. Even numbers with even abundance are in <a href="/A088829" title="Even numbers with even abundance.">A088829</a>. - <a href="/wiki/User:Jaroslav_Krizek">Jaroslav Krizek</a>, May 07 2011</div> <div class=sectline>Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_{k>=1} (1-2*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment in <a href="/A007509" title="Numerator of Sum_{k=0..n} (-1)^k/(2*k+1).">A007509</a>. - <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, Oct 12 2011</div> <div class=sectline>Ulam's spiral (SE spoke). - <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Oct 31 2011</div> <div class=sectline>All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Mar 19 2012</div> <div class=sectline>Right edge of both triangles <a href="/A214604" title="Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.">A214604</a> and <a href="/A214661" title="Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k...">A214661</a>: a(n) = <a href="/A214604" title="Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.">A214604</a>(n+1,n+1) = <a href="/A214661" title="Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k...">A214661</a>(n+1,n+1). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 25 2012</div> <div class=sectline>Also: Odd numbers which have an odd sum of divisors (= sigma = <a href="/A000203" title="a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).">A000203</a>). - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Feb 23 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 even leg b (<a href="/A231100" title="Even legs of primitive Pythagorean triples (with multiplicity) sorted with respect to increasing hypotenuse.">A231100</a>); sequence gives values c-b, sorted with duplicates removed. - <a href="/wiki/User:K._G._Stier">K. G. Stier</a>, Nov 04 2013</div> <div class=sectline>For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n-2)*(n-1),(n-1)*n/2), ((n-1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2). - <a href="/wiki/User:J._M._Bergot">J. M. Bergot</a>, May 27 2014</div> <div class=sectline>Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. - <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Nov 28 2014</div> <div class=sectline>Except for a(1)=4, the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - <a href="/wiki/User:Robert_Price">Robert Price</a>, May 23 2016</div> <div class=sectline>a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. - <a href="/wiki/User:Ivan_N._Ianakiev">Ivan N. Ianakiev</a>, Dec 21 2016</div> <div class=sectline>a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent. - <a href="/wiki/User:Indranil_Ghosh">Indranil Ghosh</a>, Dec 25 2016</div> <div class=sectline>Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is <a href="/A197037" title="Decimal expansion of the modified Struve L-function of order 0 at 1.">A197037</a>. - <a href="/wiki/User:Benedict_W._J._Irwin">Benedict W. J. Irwin</a>, Jun 21 2018</div> <div class=sectline>Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/<a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>(n), q = <a href="/A060300" title="a(n) = (2n(n+1))^2.">A060300</a>(n)/<a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>(n) = <a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>(n) - p} and their ratio p/q = a(n)/<a href="/A060300" title="a(n) = (2n(n+1))^2.">A060300</a>(n) are irreducible fractions in Q\Z. X values are <a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>, Y values are <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>, Z values are <a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>. - <a href="/wiki/User:Ralf_Steiner">Ralf Steiner</a>, Feb 25 2020</div> <div class=sectline>a(n) is the number of large or small squares that are used to tile primitive squares of type 2 (<a href="/A344332" title="Side s of squares of type 2 that can be tiled with squares of two different sizes so that the number of large or small squar...">A344332</a>). - <a href="/wiki/User:Bernard_Schott">Bernard Schott</a>, Jun 03 2021</div> <div class=sectline>Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', see <a href="/A348005" title="Positive even integers with an even number of even divisors.">A348005</a>). - <a href="/wiki/User:Bernard_Schott">Bernard Schott</a>, Nov 21 2021</div> <div class=sectline>a(n) is the least odd number k = x + y, with 0 < x < y, such that there are n distinct pairs (x,y) for which x*y/k is an integer; for example, a(2) = 25 and the two corresponding pairs are (5,20) and (10,15). The similar sequence with 'even' is <a href="/A016742" title="Even squares: a(n) = (2*n)^2.">A016742</a> (see Comment of Jan 26 2018). - <a href="/wiki/User:Bernard_Schott">Bernard Schott</a>, Feb 24 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Paolo Xausa, <a href="/A016754/b016754.txt">Table of n, a(n) for n = 0..9999</a> (terms 0..1000 from T. D. Noe)</div> <div class=sectline>Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://arxiv.org/abs/1810.07895">Classes of Gap Balancing Numbers</a>, arXiv:1810.07895 [math.NT], 2018.</div> <div class=sectline>Bruce C. Berndt and Ken Ono, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s42berndt.html">Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary</a>, S茅minaire Lotharingien de Combinatoire, B42c (1999), 63 pp.</div> <div class=sectline>John Elias, <a href="/A016754/a016754.png">Illustration: 8-fold Square Progression of Ulam's Spiral</a>.</div> <div class=sectline>Milan Janjic, <a href="https://pmf.unibl.org/janjic/">Two Enumerative Functions</a>; also on <a href="https://www.semanticscholar.org/paper/Two-Enumerative-Functions-Janjic/801b6b226bfe1d6b002fb4946c3957d7052132bd?p2df">Semantic Scholar</a>.</div> <div class=sectline>Scientific American, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue</a>.</div> <div class=sectline>Amelia Carolina Sparavigna, <a href="http://doi.org/10.5281/zenodo.3247003">Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers)</a>, Politecnico di Torino (Italy, 2019).</div> <div class=sectline>Leo Tavares, <a href="/A016754/a016754.jpg">Illustration: Diamond Triangles</a>.</div> <div class=sectline>Leo Tavares, <a href="/A016754/a016754_1.jpg">Illustration: Diamond Stars</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MooreNeighborhood.html">Moore Neighborhood</a>.</div> <div class=sectline>R. Yin, J. Mu, and T. Komatsu, <a href="https://doi.org/10.20944/preprints202407.2280.v1">The p-Frobenius Number for the Triple of the Generalized Star Numbers</a>, Preprints 2024, 2024072280. See p. 2.</div> <div class=sectline><a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</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 + Sum_{i=1..n} 8*i = 1 + 8*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n). - Xavier Acloque, Jan 21 2003; <a href="/wiki/User:Zak_Seidov">Zak Seidov</a>, May 07 2006; <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Dec 29 2010</div> <div class=sectline>O.g.f.: (1+6*x+x^2)/(1-x)^3. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jan 11 2008</div> <div class=sectline>a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1. - <a href="/wiki/User:Artur_Jasinski">Artur Jasinski</a>, Mar 27 2008</div> <div class=sectline>a(n) = <a href="/A061038" title="Denominator of 1/4 - 1/n^2.">A061038</a>(2+4n). - <a href="/wiki/User:Paul_Curtz">Paul Curtz</a>, Oct 26 2008</div> <div class=sectline>Sum_{n>=0} 1/a(n) = Pi^2/8 = <a href="/A111003" title="Decimal expansion of Pi^2/8.">A111003</a>. - <a href="/wiki/User:Jaume_Oliver_Lafont">Jaume Oliver Lafont</a>, Mar 07 2009</div> <div class=sectline>a(n) = <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>(<a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>(n)). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 08 2009</div> <div class=sectline>a(n) = a(n-1) + 8*n with n>0, a(0)=1. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Aug 01 2010</div> <div class=sectline>a(n) = <a href="/A033951" title="Write 1,2,... in a clockwise spiral; sequence gives numbers on positive x axis.">A033951</a>(n) + n. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, May 17 2009</div> <div class=sectline>a(n) = <a href="/A033996" title="8 times triangular numbers: a(n) = 4*n*(n+1).">A033996</a>(n) + 1. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Oct 03 2011</div> <div class=sectline>a(n) = (<a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>(n))^2. - <a href="/wiki/User:Zak_Seidov">Zak Seidov</a>, Nov 29 2011</div> <div class=sectline>From <a href="/wiki/User:George_F._Johnson">George F. Johnson</a>, Sep 05 2012: (Start)</div> <div class=sectline>a(n+1) = a(n) + 4 + 4*sqrt(a(n)).</div> <div class=sectline>a(n-1) = a(n) + 4 - 4*sqrt(a(n)).</div> <div class=sectline>a(n+1) = 2*a(n) - a(n-1) + 8.</div> <div class=sectline>a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2).</div> <div class=sectline>(a(n+1) - a(n-1))/8 = sqrt(a(n)).</div> <div class=sectline>a(n+1)*a(n-1) = (a(n)-4)^2.</div> <div class=sectline>a(n) = 2*<a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n) + 1 = 2*<a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>(n) - 1 = <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n) + <a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>(n).</div> <div class=sectline>Limit_{n -> oo} a(n)/a(n-1) = 1. (End)</div> <div class=sectline>a(n) = binomial(2*n+2,2) + binomial(2*n+1,2). - <a href="/wiki/User:John_Molokach">John Molokach</a>, Jul 12 2013</div> <div class=sectline>E.g.f.: (1 + 8*x + 4*x^2)*exp(x). - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, May 23 2016</div> <div class=sectline>a(n) = <a href="/A101321" title="Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.">A101321</a>(8,n). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 28 2016</div> <div class=sectline>Product_{n>=1} <a href="/A033996" title="8 times triangular numbers: a(n) = 4*n*(n+1).">A033996</a>(n)/a(n) = Pi/4. - <a href="/wiki/User:Daniel_Suteu">Daniel Suteu</a>, Dec 25 2016</div> <div class=sectline>a(n) = <a href="/A014105" title="Second hexagonal numbers: a(n) = n*(2*n + 1).">A014105</a>(n) + <a href="/A000384" title="Hexagonal numbers: a(n) = n*(2*n-1).">A000384</a>(n+1). - <a href="/wiki/User:Bruce_J._Nicholson">Bruce J. Nicholson</a>, Nov 11 2017</div> <div class=sectline>a(n) = <a href="/A003215" title="Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).">A003215</a>(n) + <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n). - <a href="/wiki/User:Klaus_Purath">Klaus Purath</a>, Jun 09 2020</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jun 20 2020: (Start)</div> <div class=sectline>Sum_{n>=0} a(n)/n! = 13*e.</div> <div class=sectline>Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/e. (End)</div> <div class=sectline>Sum_{n>=0} (-1)^n/a(n) = <a href="/A006752" title="Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...">A006752</a>. - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Oct 10 2020</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jan 28 2021: (Start)</div> <div class=sectline>Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2).</div> <div class=sectline>Product_{n>=1} (1 - 1/a(n)) = Pi/4 (<a href="/A003881" title="Decimal expansion of Pi/4.">A003881</a>). (End)</div> <div class=sectline>From <a href="/wiki/User:Leo_Tavares">Leo Tavares</a>, Nov 24 2021: (Start)</div> <div class=sectline>a(n) = <a href="/A014634" title="a(n) = (2*n+1)*(4*n+1).">A014634</a>(n) - <a href="/A002943" title="a(n) = 2*n*(2*n+1).">A002943</a>(n). See Diamond Triangles illustration.</div> <div class=sectline>a(n) = <a href="/A003154" title="Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.">A003154</a>(n+1) - <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n). See Diamond Stars illustration. (End)</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Mar 11 2024: (Start)</div> <div class=sectline>Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 ))))).</div> <div class=sectline>3/2 - 2*log(2) = Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 1/(9 - 3/(17 - 60/(33 - 315/(57 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*2^2 - ... ))))).</div> <div class=sectline>Row 2 of <a href="/A142992" title="Square array, read by ascending antidiagonals, of the crystal ball sequences for the root lattices of type C_n.">A142992</a>. (End)</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Mar 26 2024: (Start)</div> <div class=sectline>8*a(n) = (2*n + 1)*(a(n+1) - a(n-1)).</div> <div class=sectline>Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1/2 - Pi/8 = 1/(9 + (1*3)/(8 + (3*5)/(8 + ... + (4*n^2 - 1)/(8 + ... )))). For the continued fraction use Lorentzen and Waadeland, p. 586, equation 4.7.9 with n = 1. Cf. <a href="/A057813" title="a(n) = (2*n+1)*(4*n^2+4*n+3)/3.">A057813</a>. (End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline><a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>[nmax_]:=Range[1, 2nmax+1, 2]^2; <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>[100] (* <a href="/wiki/User:Paolo_Xausa">Paolo Xausa</a>, Mar 05 2023 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n)=(n<<1+1)^2 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jun 16 2011, corrected and edited by <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Apr 11 2023</div> <div class=sectline>(Haskell)</div> <div class=sectline>a016754 n = a016754_list !! n</div> <div class=sectline>a016754_list = scanl (+) 1 $ tail a008590_list</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 02 2012</div> <div class=sectline>(Maxima) <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n):=(n+n+1)^2$</div> <div class=sectline>makelist(<a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n), n, 0, 20); /* <a href="/wiki/User:Martin_Ettl">Martin Ettl</a>, Nov 12 2012 */</div> <div class=sectline>(Magma) [n^2: n in [1..100 by 2]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jan 03 2017</div> <div class=sectline>(Python)</div> <div class=sectline>def <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n): return ((n<<1)|1)**2 # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Jul 06 2023</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="/A000384" title="Hexagonal numbers: a(n) = n*(2*n-1).">A000384</a>, <a href="/A001263" title="Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.">A001263</a>, <a href="/A001539" title="a(n) = (4*n+1)*(4*n+3).">A001539</a>, <a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>, <a href="/A003881" title="Decimal expansion of Pi/4.">A003881</a>, <a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>, <a href="/A006752" title="Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...">A006752</a>, <a href="/A014105" title="Second hexagonal numbers: a(n) = n*(2*n + 1).">A014105</a>, <a href="/A016742" title="Even squares: a(n) = (2*n)^2.">A016742</a>, <a href="/A016802" title="a(n) = (4*n)^2.">A016802</a>, <a href="/A016814" title="a(n) = (4*n + 1)^2.">A016814</a>, <a href="/A016826" title="a(n) = (4n + 2)^2.">A016826</a>, <a href="/A016838" title="a(n) = (4n + 3)^2.">A016838</a>, <a href="/A033996" title="8 times triangular numbers: a(n) = 4*n*(n+1).">A033996</a>, <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>, <a href="/A060300" title="a(n) = (2n(n+1))^2.">A060300</a>, <a href="/A138393" title="Numbers of form 8k+1 which are not squares.">A138393</a>, <a href="/A167661" title="Number of partitions of n into odd squares.">A167661</a>, <a href="/A167700" title="Number of partitions of n into distinct odd squares.">A167700</a>.</div> <div class=sectline>Cf. <a href="/A000447" title="a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.">A000447</a> (partial sums).</div> <div class=sectline>Cf. <a href="/A005917" title="Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.">A005917</a>, <a href="/A344330" title="Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the...">A344330</a>, <a href="/A344332" title="Side s of squares of type 2 that can be tiled with squares of two different sizes so that the number of large or small squar...">A344332</a>.</div> <div class=sectline>Cf. <a href="/A348005" title="Positive even integers with an even number of even divisors.">A348005</a>.</div> <div class=sectline>Partial sums of <a href="/A022144" title="Coordination sequence for root lattice B_2.">A022144</a>.</div> <div class=sectline>Sequences on the four axes of the square spiral: Starting at 0: <a href="/A001107" title="10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).">A001107</a>, <a href="/A033991" title="a(n) = n*(4*n-1).">A033991</a>, <a href="/A007742" title="a(n) = n*(4*n+1).">A007742</a>, <a href="/A033954" title="Second 10-gonal (or decagonal) numbers: n*(4*n+3).">A033954</a>; starting at 1: <a href="/A054552" title="a(n) = 4*n^2 - 3*n + 1.">A054552</a>, <a href="/A054556" title="a(n) = 4*n^2 - 9*n + 6.">A054556</a>, <a href="/A054567" title="a(n) = 4*n^2 - 7*n + 4.">A054567</a>, <a href="/A033951" title="Write 1,2,... in a clockwise spiral; sequence gives numbers on positive x axis.">A033951</a>.</div> <div class=sectline>Sequences on the four diagonals of the square spiral: Starting at 0: <a href="/A002939" title="a(n) = 2*n*(2*n-1).">A002939</a> = 2*<a href="/A000384" title="Hexagonal numbers: a(n) = n*(2*n-1).">A000384</a>, <a href="/A016742" title="Even squares: a(n) = (2*n)^2.">A016742</a> = 4*<a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A002943" title="a(n) = 2*n*(2*n+1).">A002943</a> = 2*<a href="/A014105" title="Second hexagonal numbers: a(n) = n*(2*n + 1).">A014105</a>, <a href="/A033996" title="8 times triangular numbers: a(n) = 4*n*(n+1).">A033996</a> = 8*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>; starting at 1: <a href="/A054554" title="a(n) = 4*n^2 - 10*n + 7.">A054554</a>, <a href="/A053755" title="a(n) = 4*n^2 + 1.">A053755</a>, <a href="/A054569" title="a(n) = 4*n^2 - 6*n + 3.">A054569</a>, <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>.</div> <div class=sectline>Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: <a href="/A035608" title="Expansion of g.f. x*(1 + 3*x)/((1 + x)*(1 - x)^3).">A035608</a>, <a href="/A156859" title="The main column of a version of the square spiral.">A156859</a>, <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a> = 2*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>, <a href="/A137932" title="Terms in an n X n spiral that do not lie on its principal diagonals.">A137932</a> = 4*<a href="/A002620" title="Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).">A002620</a>; starting at 1: <a href="/A317186" title="One of many square spiral sequences: a(n) = n^2 + n - floor((n-1)/2).">A317186</a>, <a href="/A267682" title="a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3, with initial terms 1, 1, 4, 8.">A267682</a>, <a href="/A002061" title="Central polygonal numbers: a(n) = n^2 - n + 1.">A002061</a>, <a href="/A080335" title="Diagonal in square spiral or maze arrangement of natural numbers.">A080335</a>.</div> <div class=sectline>Cf. <a href="/A014634" title="a(n) = (2*n+1)*(4*n+1).">A014634</a>, <a href="/A003154" title="Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.">A003154</a>.</div> <div class=sectline>Sequence in context: <a href="/A325701" title="Nonprime Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.">A325701</a> <a href="/A113745" title="Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 8 multiples of n-1...">A113745</a> <a href="/A348742" title="Odd numbers k for which A161942(k) >= k, where A161942 is the odd part of sigma.">A348742</a> * <a href="/A110487" title="Squares of the form p*q - p - q + 2, where p and q are primes.">A110487</a> <a href="/A377654" title="Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has widt...">A377654</a> <a href="/A259417" title="Even powers of the odd primes listed in increasing order.">A259417</a></div> <div class=sectline>Adjacent sequences: <a href="/A016751" title="a(n) = (2*n)^11.">A016751</a> <a href="/A016752" title="a(n) = (2*n)^12.">A016752</a> <a href="/A016753" title="Expansion of 1/((1-3*x)*(1-4*x)*(1-5*x)).">A016753</a> * <a href="/A016755" title="Odd cubes: a(n) = (2*n + 1)^3.">A016755</a> <a href="/A016756" title="a(n) = (2*n+1)^4.">A016756</a> <a href="/A016757" title="a(n) = (2*n+1)^5.">A016757</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><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>Additional description from <a href="/wiki/User:Terrel_Trotter,_Jr.">Terrel Trotter, Jr.</a>, Apr 06 2002</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 December 2 21:25 EST 2024. Contains 378341 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>