%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, <a href="b000217.txt">Table of n, a(n) for n = 0..10000</
a>
%H A000217 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000217 H. Bottomley, <a href="a2378.gif">Illustration of initial terms of A000217,
A002378</a>
%H A000217 Scott A. Brown, <a href="http://home.neo.rr.com/scottbrown">Brown's Math
Page, etc.</a>
%H A000217 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000217 S. S. Gupta, <a href="http://www.shyamsundergupta.com/triangle.htm">Fascinating
Triangular Numbers</a>
%H A000217 C. Hamberg, <a href="http://staff.imsa.edu/math/journal/volume1/articles/
Triangular.pdf">Triangular Numbers Are Everywhere</a>
%H A000217 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=253">
Encyclopedia of Combinatorial Structures 253</a>
%H A000217 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000217 R. Jovanovic, <a href="http://milan.milanovic.org/math/english/triangular/
triangular.html">Triangular numbers</a>
%H A000217 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviTroug">
First 2500 Triangular numbers</a>
%H A000217 H. K. Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
"On Regular polytope numbers"</a>.
%H A000217 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Partial Complements and Transposable Dispersions</
a>, J. Integer Seqs., Vol. 7, 2004.
%H A000217 J. Koller, <a href="http://www.mathematische-basteleien.de/triangularnumber.htm">
Triangular Numbers</a>
%H A000217 A. J. F. Leatherland, <a href="http://yoyo.cc.monash.edu.au/~bunyip/primes/
triangleUlam.htm">Triangle Numbers on Ulam Spiral</a>
%H A000217 Ivars Peterson, <a href="http://www.maa.org/mathtourist/mathtourist_11_7_07.html">
Triangular Numbers and Magic Squares</a>.
%H A000217 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000217 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000217 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000217 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica
de los numeros primos y perfectos</a>.
%H A000217 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/triangle.htm">
Triangle numbers</a>
%H A000217 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">
Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal
of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H A000217 N. J. A. Sloane, <a href="a217.gif">Illustration of initial terms of
A000217, A000290, A000326</a>
%H A000217 Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/
word/1019">Triangular Numbers</a>
%H A000217 T. Trotter, <a href="http://www.trottermath.net/numthry/trident.html">
Some Identities for the Triangular Numbers</a>, J. Rec. Math. vol.
6, no. 2 Spring 1973.
%H A000217 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
Geometri/NbTrianB.htm">Nombres Triangulaires</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TriangularNumber.html">Link to a section of The World of Mathematics
(1).</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
AbsoluteValue.html">Link to a section of The World of Mathematics
(2).</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Composition.html">Link to a section of The World of Mathematics (3).</
a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Distance.html">Link to a section of The World of Mathematics (4).</
a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GolombRuler.html">Link to a section of The World of Mathematics (5).</
a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PolygonalNumber.html">Link to a section of The World of Mathematics
(6).</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
LineLinePicking.html">Line Line Picking</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TrinomialCoefficient.html">Trinomial Coefficient</a>
%H A000217 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
WienerIndex.html">Wiener Index</a>
%H A000217 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000217 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000217 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000217 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%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,i<a,s+=i;i++ ];x=s]; [From Vladimir Orlovsky
(4vladimir(AT)gmail.com), Apr 03 2009]
%t A000217 Array[ #*(# - 1)/2 &, 54, 1] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 10 2009]
%o A000217 (PARI) a(n)=n*(n+1)/2
%o A000217 sage: [lucas_number1(3,n,n)/2 for n in xrange(1,55)] - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jul 15 2008
%o A000217 (Other) sage: [stirling_number1(n,n-1) for n in xrange(1, 55)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
%Y A000217 Cf. A007318, A002024, A000096, A000124, A002378, A000292, A000330.
%Y A000217 Cf. A000096, A055998, A055999, A056000, A056115, A056119, A056121, A056126,
A051942, A101859, A001477.
%Y A000217 Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475.
%Y A000217 Cf. A006011, A002415, A010054.
%Y A000217 a(2k-1)=A000384(k), a(2k)=A014105(k), k>0.
%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).
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