Search: id:A000217 Results 1-1 of 1 results found. %I A000217 M2535 N1002 %S A000217 0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231, %T A000217 253,276,300,325,351,378,406,435,465,496,528,561,595,630,666,703,741, %U A000217 780,820,861,903,946,990,1035,1081,1128,1176,1225,1275,1326,1378,1431 %N A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. %C A000217 Number of edges in complete graph of order n, K_n. %C A000217 Number of legal ways to insert a pair of parentheses in a string of n letters. E.g. there are 6 ways for three letters: (a)bc, (ab)c, (abc), a(b)c, a(bc), ab(c). [Proof: there are C(n+2,2) ways to choose where the parentheses might go, but n+1 of them are illegal because the parentheses are adjacent.] Cf. A002415. %C A000217 For n >= 1 a(n)=n(n+1)/2 is also the genus of a nonsingular curve of degree n+2 like the Fermat curve x^(n+2) + y^(n+2) = 1 - Ahmed Fares (ahmedfares(AT)my_deja.com), Feb 21 2001 %C A000217 From Harnack's theorem (1876), the number of branches of a non-singuliar curve of order n is bounded by a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 29 2002 %C A000217 Number of tiles in the set of double-n dominoes. - Scott A. Brown (scottbrown(AT)neo.rr.com), Sep 24 2002 %C A000217 Number of ways a chain of n non-identical links can be be broken up. This is based on a similar problem in the field of proteomics: the number of ways a peptide of n amino acid residues can be be broken up in a mass spectrometer. In general each amino acid has a different mass, so AB and BC would have different masses. - James Raymond (raymond(AT)unlv.edu), Apr 08 2003 %C A000217 Maximum number of intersections of n+1 lines which may only have 2 lines per intersection point. Maximal number of closed regions when n+1 lines are maximally 2-intersected in given by T(n-1). Using n+1 lines with k>1 parallel lines, the maximum number of 2-intersections is given by T(n)-T(k-1). - Jon Perry (perry(AT)globalnet.co.uk), Jun 11 2003 %C A000217 Number of distinct straight lines that can pass through n points in 3-dimensional space. - Cino Hilliard (hillcino368(AT)gmail.com), Aug 12 2003 %C A000217 Triangular numbers - odd numbers = triangular numbers: 0,1,3,6,10,15, 21... - 0,1,3,5,7,9,11... = 0,0,0,1,3,6,10... - Xavier Acloque Oct 31 2003 %C A000217 Centered polygonal numbers are the result of [number of sides * A000217 + 1]. E.g. centered pentagonal numbers (1,6,16,31...)= 5 * (0,1,3, 6...) + 1. Centered heptagonal numbers (1,8,22,43...)= 7 * (0,1,3, 6...) + 1. - Xavier Acloque Oct 31 2003 %C A000217 Maximum number of lines formed by the intersection of n+1 planes. - Ronald R. King (king_ron(AT)asdk12.org), Mar 29 2004 %C A000217 Number of permutations of [n] which avoid the pattern 132 and have exactly 1 descent. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004 %C A000217 a(n) == 1 mod (n+2) if n is odd and == n/2+2 mod (n+2) if n is even. - Jon Perry (perry(AT)globalnet.co.uk), Dec 16 2004 %C A000217 Number of ways two different numbers can be selected from the set {0, 1,2,...,n} without repetition, or, number of ways two different numbers can be selected from the set {1,2,...,n} with repetition. %C A000217 1, 6, 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005 %C A000217 a(n) = A108299(n+3,4) = -A108299(n+4,5). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005 %C A000217 In 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n=256,..,511, the number of non-color partitions are computable with A045943(n-255), while for n = 512-765, the number of color points in r+g+b planes equals A000217(765-n). - Labos E. (labos(AT)ana.sote.hu), Jun 20 2005 %C A000217 A110560/A110561 = numerator/denominator of the coefficients of the exponential generating function. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 27 2005 %C A000217 Binomial transform is {0, 1, 5, 18, 56, 160, 432, ... }, A001793 with one leading zero . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2005 %C A000217 a(n) = A111808(n,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005 %C A000217 Each pair of neighboring terms adds to a perfect square. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 21 2006 %C A000217 a(n)*a(n+1) = A006011(n) = n^2*(n^2-1)/4 = 3*A002415(n) = 1/2*a(n^2+2*n). a(n-1)*a(n) = 1/2*a(n^2-1). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006 %C A000217 Number of transpositions in the symmetric group of n+1 letters i.e. the number of permutations that leave all but two elements fixed. - Geoffrey Critzer (geoffreycritzer(AT)yahoo.com), Jun 23 2006 %C A000217 Beginning from a(3), a(n) is the number of way to get a semiprime from n primes. Example: From 2 and 3 the number of semiprimes is 3: 2*2, 3*3, 2*3; from 2 and 3 and 5 the number of semiprimes is 6: 2*2, 3*3, 5*5, 2*3, 2*5, 3*5. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Sep 17 2006 %C A000217 With rho(n):=exp(i*2*Pi/n) (an n-th root of 1) one has, for n>=1, rho(n)^a(n)=(-1)^(n+1). Just use the triviality a(2*k+1)=0(mod (2*k+1)) and a(2*k)=k(mod 2*k). %C A000217 Comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 18 2006: (Start) %C A000217 a:=n->sum(j + 1,j=-1..n): seq(a(n),n=-1..50); %C A000217 a:=n->sum(j + 2,j=0..n): seq(a(n),n=-1..51); => A000096 = this sequence + 1*A001477 %C A000217 a:=n->sum(j + 2,j=1..n): seq(a(n),n=0..48); => A055998 = this sequence + 2*A001477 %C A000217 a:=n->sum(j + 2,j=2..n):seq(a(n),n=1..50); => A055999 = this sequence + 3*A001477 %C A000217 a:=n->sum(j + 2,j=3..n):seq(a(n),n=2..52); => A056000 = this sequence + 4*A001477 %C A000217 a:=n->sum(j + 2,j=4..n):seq(a(n),n=3..53); => A056115 = this sequence + 5*A001477 %C A000217 a:=n->sum(j + 2,j=5..n):seq(a(n),n=4..54); => A056119 = this sequence + 6*A001477 %C A000217 a:=n->sum(j + 2,j=6..n):seq(a(n),n=5..50); => A056121 = this sequence + 7*A001477 %C A000217 a:=n->sum(j + 2,j=7..n):seq(a(n),n=6..56); => A056126 = this sequence + 8*A001477 %C A000217 a:=n->sum(j + 2,j=8..n):seq(a(n),n=7..56); => A051942 = this sequence + 9*A001477 %C A000217 a:=n->sum(j + 2,j=9..n):seq(a(n),n=8..59); => A101859 = this sequence + 10*A001477 (End) %C A000217 a(n) = A126890(n,0). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006 %C A000217 a(n) is the number of terms in the expansion of (a_1+a_2+a_3)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007 %C A000217 (sqrt(8 a(n) + 1) - 1)/2 = n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007 %C A000217 The number of distinct handshakes in a room with n people (n>=2). - Mohammad K. Azarian (azarian(AT)evansville.edu), Apr 12 2007 %C A000217 Equal to the rank (minimal cardinality of a generating set) of the semigroup PT_n\S_n, where PT_n and S_n denote the partial transformation semigroup and symmetric group on [n]. - James East (james.east(AT)latrobe.edu.au), May 03 2007 %C A000217 Gives the total number of triangles found when cevians are drawn from a single vertex on a triangle to the side opposite that vertex, where n=the number of cevians drawn+1. For instance, with 1 cevian drawn, n=1+1=2 and a(n)=2(2+1)/2=3 so there is a total of 3 triangles in the figure. If 2 cevians are drawn from one point to the opposite side, then n=1+2=3 and a(n)=3(3+1)/2=6 so there is a total of 6 triangles in the figure. - Noah Priluck (npriluck(AT)gmail.com), Apr 30 2007 %C A000217 a(n), n>=1, is the number of ways in which n-1 can be written as a sum of three positive integers if representations differing in the order of the terms are considered to be different. In other words a(n), n>=1, is the number of positive integral solutions of the equation x + y + z = n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 22 2001 %C A000217 a(n+1), n>=0, is the number of levels with energy n+3/2 (in units of h*f0, with Planck's constant h and the oscillator frequency f0) of the three dimensional isotropic harmonic quantum oscillator. See the comment by A. Murthy above: n=n1+n2+n3 with positive integers and ordered. Proof from the o.g.f. See the A. Messiah reference. W. Lang, Jun 29 2007. %C A000217 Numbers m>=0 such that round(sqrt(2m+1))-round(sqrt(2m))=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007 %C A000217 Numbers m>=0 such that ceiling(2*sqrt(2m+1))-1=1+floor(2*sqrt(2m)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007 %C A000217 Numbers m>=0 such that fract(sqrt(2m+1))>1/2 and fract(sqrt(2m))<1/2, where fract(x) is the fractional part of x (i.e. x-floor(x), x>=0). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007 %C A000217 Each term, except for the initial 0, is a sum of digits of terms in A007908. - Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 01 2007 %C A000217 Sequence allows us to find X values of the equation: 8*X^3 + X^2 = Y^2. To find Y values: b(n)=n(n+1)(2n+1)/2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007 %C A000217 If Y and Z are 3-blocks of an n-set X then, for n>=6, a(n-1) is the number of (n-2)-subsets of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Nov 09 2007 %C A000217 Number of n-permutations of 2 objects u,v, with repetition allowed, containing exactly two u's. Example: n=4, a(4) = 6 because we have uuvv, uvuv, vuuv, uvvu, vuvu and vvuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 15 2008 %C A000217 Equals row sums of triangle A143320, n>0. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008] %C A000217 a(n) is also a perfect number A000396 when n is a Mersenne prime A000668. [From Omar E. Pol (info(AT)polprimos.com), Sep 05 2008] %C A000217 a(n) = A022264(n) - A049450(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 09 2008] %C A000217 Equals row sums of triangle A152204 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008] %C A000217 The number of matches played in a round robin tournament: n*(n-1)/2 gives the number of matches needed for n players. Everyone plays against everyone else exactly once. [From Georg Wrede (georg(AT)iki.fi), Dec 18 2008] %C A000217 -a(n+1) = E(2)*C(n+2,2) (n>=0) where E(n) are the Euler number in the enumeration A122045 and C(n,k) are the binomial coefficients A007318. Viewed this way a(n) is the special case k=2 in the sequence of diagonals in the triangle A153641. [From Peter Luschny (peter(AT)luschny.de), Jan 06 2009] %C A000217 4a(x)+4a(y)+1=(x+y+1)^2+(x-y)^2 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jan 21 2009] %C A000217 Equivalent to the first differences of successive tetrahedral numbers. See A000292. [From Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009] %C A000217 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start) %C A000217 The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n, 2k+1)). Thus %C A000217 a(k) = |2^(-3)(P(2,1)-(-1)^k P(2,2k+1))|. (End) %C A000217 a(n) is the smallest number > a(n-1) such that gcd(n,a(n)) = gcd(n,a(n-1)). If n is odd this gcd is n; if n is even it is n/2. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Aug 06 2009] %C A000217 Number of units of a(n) belongs to a periodic sequence: 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009] %C A000217 a(A006894(n)) = a(A072638(n-1)+1) = A072638(n) = A006894(n+1)-1 for n >= 1. For n=4, a(11) = 66. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 12 2009] %D A000217 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828. %D A000217 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2. %D A000217 Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5. %D A000217 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. %D A000217 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 109ff. %D A000217 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155. %D A000217 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1. %D A000217 Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8. %D A000217 Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4. %D A000217 Labos E.: On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06,2005. %D A000217 A. Messiah, Quantum Mechanics, Vol.1, North Holland, Amsterdam, 1965, p. 457. %D A000217 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954. %D A000217 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000217 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000217 T. Trotter, Some Identities for the Triangular Numbers, Journal of Recreational Mathematics, Spring 1973, 6(2). %D A000217 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 91-3 Penguin Books 1987. %D A000217 Michael Boardman, "The Egg-Drop Numbers", Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Sep 30 2009] %H A000217 N. J. A. Sloane, Table of n, a(n) for n = 0..10000 %H A000217 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. %H A000217 H. Bottomley, Illustration of initial terms of A000217, A002378 %H A000217 Scott A. Brown, Brown's Math Page, etc. %H A000217 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000217 S. S. Gupta, Fascinating Triangular Numbers %H A000217 C. Hamberg, Triangular Numbers Are Everywhere %H A000217 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 253 %H A000217 Milan Janjic, Two Enumerative Functions %H A000217 R. Jovanovic, Triangular numbers %H A000217 R. Jovanovic, First 2500 Triangular numbers %H A000217 H. K. Kim, "On Regular polytope numbers". %H A000217 Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004. %H A000217 J. Koller, Triangular Numbers %H A000217 A. J. F. Leatherland, Triangle Numbers on Ulam Spiral %H A000217 Ivars Peterson, Triangular Numbers and Magic Squares. %H A000217 Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7 %H A000217 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000217 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000217 O. E. Pol, Determinacion geometrica de los numeros primos y perfectos. %H A000217 F. Richman, Triangle numbers %H A000217 James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4. %H A000217 N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326 %H A000217 Thesaurus.maths.org, Triangular Numbers %H A000217 T. Trotter, Some Identities for the Triangular Numbers, J. Rec. Math. vol. 6, no. 2 Spring 1973. %H A000217 G. Villemin's Almanach of Numbers, Nombres Triangulaires %H A000217 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1). %H A000217 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2). %H A000217 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3). %H A000217 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (4). %H A000217 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (5). %H A000217 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (6). %H A000217 Eric Weisstein's World of Mathematics, Line Line Picking %H A000217 Eric Weisstein's World of Mathematics, Trinomial Coefficient %H A000217 Eric Weisstein's World of Mathematics, Wiener Index %H A000217 Index entries for "core" sequences %H A000217 Index entries for related partition-counting sequences %H A000217 Index entries for two-way infinite sequences %H A000217 Index entries for sequences related to linear recurrences with constant coefficients %F A000217 G.f.: x/(1-x)^3. E.g.f.: exp(x)(x+x^2/2). a(n)=a(-1-n). %F A000217 a(n) = a(n-1)+n. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 06 2005 %F A000217 a(n) + a(n-1)*a(n+1) = a(n)^2. - Terry Trotter (ttrotter(AT)telesal.net), Apr 08, 2002 %F A000217 a(n) = (-1)^n*sum(k=1, n, (-1)^k*k^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 29 2002 %F A000217 a(n) = ((n+2)/n)*a(n-1) %F A000217 Sum(n=1..infinity, 1/a(n)) = 2. - Jon Perry (perry(AT)globalnet.co.uk), Jul 13 2003 %F A000217 For n>0, a(n)=A001109(n)-(sum_{k=0...n-1}((2k+1)*A001652(n-1-k))) e.g. 10=204-(1*119+3*20+5*3+7*0) - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003 %F A000217 G.f.: x/(1-x)^3. E.g.f.: exp(x)(x+x^2/2). a(n)=a(-1-n). %F A000217 With interpolated zeros, this is n(n+2)/8*(1+(-1)^n)/2=sum{k=0..n, sum{j=0..k, floor(k^2/4)}}. - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003 %F A000217 a(n+1) is the determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+1, i) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003 %F A000217 a(n)=[(n^3-(n-1)^3)-(n^1-(n-1)^1)]/(2^3-2^1)= (n^3-(n-1)^3-1)/6 - Xavier Acloque Oct 24 2003 %F A000217 a(n) = a(n-1) + (1 + sqrt[1 + 8*a(n-1)])/2. E.g. a(4) = a(3) + (1 + sqrt[1 + 8*a(3)])/2 = 6 + (1 + sqrt[49])/2 = 6+8/2 = 10. This recursive relation is inverted when taking the negative branch of the square root, i.e. a(n) is transformed into a(n-1) rather than a(n+1). - Carl R. White (cyrek(AT)cyreksoft.yorks.com), Nov 04 2003 %F A000217 a(n)+a(n+1) = (n+1)^2. %F A000217 a(n) = a(n-2)+2n-1. - Paul Barry (pbarry(AT)wit.ie), Jul 17 2004 %F A000217 a(n) = Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004 %F A000217 a(n) = Sqrt[Sqrt[Sum[Sum[(i*j)^3, {i, 1, n}], {j, 1, n}]]]. a(n) = Sum[Sum[Sum[(i*j*k)^3, {i, 1, n}], {j, 1, n}], {k, 1, n}]^(1/6) - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004 %F A000217 a(0) = 0, a(1) = 1, a(n) = 2*a(n-1)-a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005 %F A000217 a(n) = Sum_{k = 1...n} phi(k)*floor(n/k) = Sum{k = 1...n} A000010(k)*A010766(n, k) (R. Dedekind). - Vladeta Jovovic = (vladeta(AT)eunet.rs), Feb 05 2004 %F A000217 a(n) = floor((2n+1)^2/8) - Paul Barry (pbarry(AT)wit.ie), May 29 2006 %F A000217 For positive n, we have a(8*a(n))/a(n) = 4*(2n+1)^2 = (4n+2)^2, i.e., a(A033996(n))/a(n) = 4*A016754(n) = (A016825(n))^2 = A016826(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 29 2006 %F A000217 [a(n)]^2+[a(n+1)]^2 = a((n+1)^2) [R B Nelsen, Math Mag 70 (2) (1997) p 130]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2006 %F A000217 a(n) = A023896(n) + A067392(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 02 2007 %F A000217 a(n)=sum(sum(j-k, j=1..n),k=0..n), n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007 %F A000217 Sum_{k, 0<=k<=n}a(k)*A039599(n,k)=A002457(n-1), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2007 %F A000217 a(n) = (n+1)^2 - a(n+1) - Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 21 2008 %F A000217 A general formula for polygonal numbers is: P(k,n) = (k-2)(n-1)n/2 + n, where P(k,n) is the n-th k-gonal number. - Omar E. Pol (info(AT)polprimos.com), Apr 28 2008 %F A000217 If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j, j=0..k-1),k=0..n-i), then a(n)=-f(n,n-1,1), for n>=1. [From Milan R. Janjic (agnus(AT)blic.net), Dec 20 2008] %F A000217 (2n+1)^2=8*a(n)+1 [From S. Vinatier (stephane.vinatier(AT)unilim.fr), Apr 03 2009] %F A000217 a(n) = A000124(n-1) + (n-1) for n >= 2. a(n) = A000124(n) - 1. A000124(n) = central polygonal numbers. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 16 2009] %F A000217 An exponential generating function for the inverse of this sequence is given by sum((pochhammer(1, m)*pochhammer(1, m))*x^m/(pochhammer(3, m)*factorial(m)), m = 0 .. infinity)=((2-2*x)*ln(1-x)+2*x)/x^2; The n-th derivative of which has a closed form which must be evaluated by taking the limit x=0. A000217[n+1]=limit(Diff(((2-2*x)*ln(1-x)+2*x)/ x^2, x$n),x=0)^-1=limit((2*GAMMA(n)*(-1/x)^n*(n*(x/(-1+x))^n*(-x+1+n)*LerchPhi(x/ (-1+x), 1, n)+(-1+x)*(n+1)*(x/(-1+x))^n+n*(ln(1-x)+ln(-1/(-1+x)))*(-x+1+n))/ x^2),x=0)^-1 [From Stephen Crowley (crow(AT)crowlogic.net), Jun 28 2009] %F A000217 a(n) = A034856(n+1) - A005408(n) = A005843(n) + A000124(n) - A005408(n) = A000124(n) - 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 05 2009] %e A000217 When n=3, a(3) = 4*3/2 = 6. %e A000217 Example(a(4)=10): ABCD where A, B, C and D are different links in a chain or different amino acids in a peptide possible fragments: A, B, C, D, AB, ABC, ABCD, BC, BCD, CD = 10 %p A000217 A000217 := proc(n) n*(n+1)/2; end; [ seq(n*(n+1)/2, n=0..100)]; %p A000217 istriangular:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/ 2 then RETURN(true) else RETURN(false); fi; end; (N. J. A. Sloane, May 25 2008) %p A000217 a[0]:=0: a[1]:=1: for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+1 od: seq(a[n], n=0..50); (Kristof) %p A000217 [seq (stirling2(n+1,n),n=1..53)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo), Dec 06 2006 %p A000217 ZL := [S, {S=Prod(B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=2..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007 %p A000217 a:=n->sum(n+2*j, j=0..n)/4: seq(a(n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2007 %p A000217 a:=n->sum(sum(j-k, j=1..n),k=0..n): seq(a(n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007 %p A000217 seq(sum(mul(gcd(j,k),j=0..n), k=0..n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007 %p A000217 A000217:=-1/(z-1)**3; [S. Plouffe in his 1992 dissertation.] %p A000217 seq(sum(binomial(n,k+1),k=1..1),n=1..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007 %p A000217 with(combinat):a:=n->sum(fibonacci(2,i), i=0..n): seq(a(n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008 %p A000217 a:=n->sum(1+sum(1, k=1..n),k=1..n):seq(a(n)/2, n=0...55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008 %p A000217 with(finance):seq(add(futurevalue(k,3,2),k=0..n)/16,n=0..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008 %p A000217 restart: G(x):=x^2*exp(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/2,n=1..54);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009] %t A000217 Table[Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]], {n, 0, 10}] %t A000217 Table[(m^2 - m)/2, {m, 54}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007 %t A000217 Table[Sqrt[StirlingS2[i+1, i]*(-StirlingS1[i+1, i])], {i,0, 53}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007 %t A000217 ...and/or... i=0;s=0;lst={};Do[s+=n+i;If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 1}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 29 2008] %t A000217 tr[a_]:=Module[{x},s=0;For[i=1,i0. %Y A000217 A diagonal of A008291. a(n) = A110555(n+2, 2). %Y A000217 a(n) = A110449(n, 0). %Y A000217 A143320 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008] %Y A000217 Cf. A000396, A000668. [From Omar E. Pol (info(AT)polprimos.com), Sep 05 2008] %Y A000217 a(3*n)=A081266, a(4*n)=A033585, a(5*n)=A144312, a(6*n)=A144314. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 17 2008] %Y A000217 A152204 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008] %Y A000217 Sequence in context: A105339 A089594 A161680 this_sequence A105340 A109811 A025747 %Y A000217 Adjacent sequences: A000214 A000215 A000216 this_sequence A000218 A000219 A000220 %K A000217 nonn,core,easy,nice %O A000217 0,3 %A A000217 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.010 seconds