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A000217
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Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
(Formerly M2535 N1002)
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2208
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0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431
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OFFSET
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0,3
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COMMENTS
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Also generalized hexagonal numbers: n*(2*n-1), n=0, +-1, +-2, +-3,... Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. In this case k = 6. - Omar E. Pol, Sep 13 2011 and Aug 04 2012
Number of edges in complete graph of order n, K_n.
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.
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
From Harnack's theorem (1876), the number of branches of a nonsingular curve of order n is bounded by a(n) - Benoit Cloitre, Aug 29 2002
Number of tiles in the set of double-n dominoes. - Scott A. Brown (scottbrown(AT)neo.rr.com), Sep 24 2002
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 A. Raymond, Apr 08 2003
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, Jun 11 2003
Number of distinct straight lines that can pass through n points in 3-dimensional space. - Cino Hilliard (hillcino368(AT)gmail.com), Aug 12 2003
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
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
Maximum number of lines formed by the intersection of n+1 planes. - Ron R. King, Mar 29 2004
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
Number of ternary words of length n-1 with subwords (0,1), (0,2) and (1,2) not allowed. - Olivier Gérard, Aug 28 2012
a(n) == 1 mod (n+2) if n is odd and == n/2+2 mod (n+2) if n is even. - Jon Perry, Dec 16 2004
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.
Conjecturally, 1, 6, 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
A110560/A110561 = numerator/denominator of the coefficients of the exponential generating function. - Jonathan Vos Post, Jul 27 2005
Binomial transform is {0, 1, 5, 18, 56, 160, 432, ... }, A001793 with one leading zero . - Philippe DELEHAM, Aug 02 2005
a(n) = A111808(n,2) for n>1. - Reinhard Zumkeller, Aug 17 2005
Each pair of neighboring terms adds to a perfect square. - Zak Seidov, Mar 21 2006
a(n)*a(n+1) = A006011(n+1) = (n+1)^2*(n^2+2)/4 = 3*A002415(n+1) = 1/2*a(n^2+2*n). a(n-1)*a(n) = 1/2*a(n^2-1). - Alexander Adamchuk, Apr 13 2006 Corrected and edited by Charlie Marion, Nov 26 2010
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, Jun 23 2006
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).
a(n) = A126890(n,0). - Reinhard Zumkeller, Dec 30 2006
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
(sqrt(8 a(n) + 1) - 1)/2 = n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007
The number of distinct handshakes in a room with n people (n>=2). - Mohammad K. Azarian, Apr 12 2007
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
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
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
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. Wolfdieter Lang, Jun 29 2007.
Numbers m>=0 such that round(sqrt(2m+1))-round(sqrt(2m))=1. - Hieronymus Fischer, Aug 06 2007
Numbers m>=0 such that ceiling(2*sqrt(2m+1))-1=1+floor(2*sqrt(2m)). - Hieronymus Fischer, Aug 06 2007
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, Aug 06 2007
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
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 Janjic, Nov 09 2007
Equals row sums of triangle A143320, n>0. [From Gary W. Adamson, Aug 07 2008]
a(n) is also a perfect number A000396 if n is a Mersenne prime A000668, assuming there are no odd perfect numbers. [From Omar E. Pol, Sep 05 2008]
a(n) = A022264(n) - A049450(n). [From Reinhard Zumkeller, Oct 09 2008]
Equals row sums of triangle A152204 [From Gary W. Adamson, Nov 29 2008]
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]
-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, Jan 06 2009]
4a(x)+4a(y)+1=(x+y+1)^2+(x-y)^2 [From Vladimir Shevelev , Jan 21 2009]
Equivalent to the first differences of successive tetrahedral numbers. See A000292. [From Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009]
Contribution from Peter Luschny, Jul 12 2009: (Start)
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
a(k) = |2^(-3)(P(2,1)-(-1)^k P(2,2k+1))|. (End)
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, Aug 06 2009]
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, Sep 12 2009]
Partial sums of A001477. [From Juri-Stepan Gerasimov, Jan 25 2010]. A-number corrected by Omar E. Pol, Jun 05 2012
The numbers along the right edge of Floyd's triangle are 1, 3, 6, 10, 15, .... [From Paul Muljadi, Jan 25 2010]
a(n)+2a(n-1)+a(n-2) = n^2+(n-1)^2; and
a(n)+3a(n-1)+3a(n-2)+a(n-3)= n^2+2*(n-1)^2+(n-2)^2.
In general, for n>=m>2,sum_{k=0,...,m}binomial(m,m-k)*a(n-k)=
sum_{k=0,...,m-1}binomial(m-1,m-1-k)*(n-k)^2.
a(n) - 2a(n-1) + a(n-2) = 1, a(n) - 3a(n-1) + 3a(n-2) - a(n-3) = 0 and
a(n) - 4a(n-1) + 6a(n-2) - 4(a-3) + a(n-4) = 0.
In general, for n>=m>2, sum_{k=0,...,m}(-1)^k*binomial(m,m-k)*a(n-k)=0.
- Charlie Marion, Oct 15 2010
More generally, a(2k+1) == j(2j-1) mod (2k+2j+1) and
a(2k) == [-k + 2j(j-1)] mod (2k+2j)
Column sums of:
1 3 5 7 9...
1 3 5...
1...
..............
--------------
1 3 6 10 15...
Sum(n=1..infinity, 1/a(n)^2)=4*pi^2/3-12 = 12 less than the volume of a sphere with radius pi^(1/3). - Charlie Marion, Dec 03 2010
A004201(a(n)) = A000290(n); A004202(a(n)) = A002378(n). [Reinhard Zumkeller, Feb 12 2011]
1/a(n+1), n>=0, has e.g.f. -2*(1+x-exp(x))/x^2, and o.g.f. 2*(x+(1-x)*ln(1-x))/x^2 (see the Stephen Crowley formula line).
-1/(2*a(n+1)) is the z-sequence for the Sheffer triangle of the coefficients of the Bernoulli polynomials A196838/A196839. [Wolfdieter Lang, Oct 26 2011]
Contribution from Charlie Marion, Feb 23 2012: (Start)
a(n) + a(A002315(k)*n+A001108(k+1)) = (A001653(k+1)*n+A001109(k+1))^2. For k=0 we obtain a(n)+a(n+1) = (n+1)^2 (identity added by N. J. A. Sloane on Feb 19 2004).
a(n) + a(A002315(k)*n-A055997(k+1)) = (A001653(k+1)*n-A001109(k))^2.
(End)
Plot the three points (0,0), (a(n), a(n+1)), (a(n+1), a(n+2)) to form a triangle. The area will be a(n+1)/2. - J. M. Bergot, May 04 2012
The sum of four consecutive triangular numbers, beginning with a(n)=n*(n+1)/2, minus 2 is 2*(n+2)^2. a(n)*a(n+2)/2 = a(a(n+1)-1). - J. M. Bergot, May 17 2012
(a(n)*a(n+3) -a(n+1)*a(n+2))*(a(n+1)*a(n+4) -a(n+2)*a(n+3))/8 = a((n^2+5*n+4)/2). - J. M. Bergot, May 18 2012
a(n)*a(n+1) + a(n+2)*a(n+3) + 3 = a(n^2 + 4*n + 6). - J. M. Bergot, May 22 2012
In general, a(n)*a(n+1) + a(n+k)*a(n+k+1) + a(k-1)*a(k) =
a(n^2 + (k+2)*n + k*(k+1)). - Charlie Marion, Sep 11 2012
a(n)*a(n+3) + a(n+1)*a(n+2) = a(n^2 + 4*n + 2). - J. M. Bergot, May 22 2012
In general, a(n)*a(n+k) + a(n+1)*a(n+k-1) = a(n^2 + (k+1)*n + k-1). - Charlie Marion, Sep 11 2012
a(n)*a(n+2) + a(n+1)*a(n+3) = a(n^2 + 4*n + 3). - J. M. Bergot, May 22 2012
Three points (a(n),a(n+1)), (a(n+1),a(n)) and (a(n+2),a(n+3)) form a triangle with area 4*a(n+1). - J. M. Bergot, May 23 2012
a(n) + a(n+k) = (n+k)^2 - (k^2 + (2n-1)*k -2n)/2. For k=1 we obtain a(n) + a(n+1) = (n+1)^2 (see below). - Charlie Marion, Oct 02 2012
In n-space we can define a(n-1) nontrivial orthogonal projections. For example, in 3-space there are a(2)=3 (namely point onto line, point onto plane, line onto plane). - Douglas Latimer, Dec 17 2012
From James East, Jan 08 2013 (Start):
For n>=1, a(n) is equal to the rank (minimal cardinality of a generating set) and idempotent rank (minimal cardinality of an idempotent generating set) of the semigroup P_n\S_n, where P_n and S_n denote the parition monoid and symmetric group on [n].
For n>=3, a(n-1) is equal to the rank and idempotent rank of the semigroup T_n\S_n, where T_n and S_n denote the full transformation semigroup and symmetric group on [n].
(End)
For n>=3, a(n) is equal to the rank and idempotent rank 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, Jan 15 2013
Conjecture: For n>0, there is always a prime between A000217(n) and A000217(n+1). Sequence A065383 has the first 1000 of these primes. - Ivan N. Ianakiev, Mar 11 2013
The formula, a(n)*a(n+4k+2)/2 + a(k) = a(a(n+2k+1)-(k^2+(k+1)^2)), is a generalization of the formula a(n)*a(n+2)/2 = a(a(n+1)-1) in Bergot's comment dated May 17 2012. [Charlie Marion, Mar 28 2013]
The series sum(1/a(k),k=1..infinity) = 2, given in a formula below by John Perry, Jul 13 2003, has partial sums 2*n/(n+1) (telescopic sum) = A022998(n)/A026741(n+1). - Wolfdieter Lang, Apr 09 2013
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REFERENCES
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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.
C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See Chapter 1.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 109ff.
Michael Boardman, "The Egg-Drop Numbers", Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi, Sep 30 2009]
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
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.
Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.
J. East, Presentations for singular subsemigroups of the partial transformation semigroup. Internat. J. Algebra Comput., 20 (2010), no. 1, 1-25.
J. East, On the singular part of the partition monoid, Internat. J. Algebra Comput. 21 (2011), no. 1-2, 147-178.
James Gleick, The Information: A History, A Theory, A Flood, Pantheon, 2011.[On page 82 mentions a table of the first 19999 triangular numbers published by E. de Joncort in 1762.]
J. M. Howie, Idempotent generators in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh Sect. A, 81 (1978), no. 3-4, 317-323.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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.
A. Messiah, Quantum Mechanics, Vol.1, North Holland, Amsterdam, 1965, p. 457.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in Combinatorial Algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. Trotter, Some Identities for the Triangular Numbers, Journal of Recreational Mathematics, Spring 1973, 6(2).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 91-3 Penguin Books 1987.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..30000
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].
T. Beldon and T. Gardiner, Triangular numbers and perfect squares, The Mathematical Gazette 86 (2002), 423-431.
Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 7-9.
H. Bottomley, Illustration of initial terms of A000217, A002378
Scott A. Brown, Brown's Math Page, etc. [Broken link?]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. S. Gupta, Fascinating Triangular Numbers
Guo-Niu Han, Enumeration of Standard Puzzles
C. Hamberg, Triangular Numbers Are Everywhere
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 35. Book's website
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 253
Milan Janjic, Two Enumerative Functions
R. Jovanovic, Triangular numbers
R. Jovanovic, First 2500 Triangular numbers
H. K. Kim, "On Regular polytope numbers".
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
J. Koller, Triangular Numbers
A. J. F. Leatherland, Triangle Numbers on Ulam Spiral
Ivars Peterson, Triangular Numbers and Magic Squares.
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
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567
F. Richman, Triangle numbers
sci.math, Jun 98 Square numbers that are triangular
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326
T. Trotter, Some Identities for the Triangular Numbers, J. Rec. Math. vol. 6, no. 2 Spring 1973.
G. Villemin's Almanach of Numbers, Nombres Triangulaires
Eric Weisstein's World of Mathematics, Triangular Number
Eric Weisstein's World of Mathematics, Absolute Value
Eric Weisstein's World of Mathematics, Composition
Eric Weisstein's World of Mathematics, Distance
Eric Weisstein's World of Mathematics, Golomb Ruler
Eric Weisstein's World of Mathematics, Polygonal Number
Eric Weisstein's World of Mathematics, Line Line Picking
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Eric Weisstein's World of Mathematics, Wiener Index
Wikipedia, Floyd's triangle
Index entries for "core" sequences
Index entries for related partition-counting sequences
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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G.f.: x/(1-x)^3.
E.g.f.: exp(x)*(x+x^2/2).
a(n) = a(-1-n).
a(n) = a(n-1)+n. - Zak Seidov, Mar 06 2005
a(n) + a(n-1)*a(n+1) = a(n)^2. - Terry Trotter (ttrotter(AT)telesal.net), Apr 08, 2002
a(n) = (-1)^n*sum(k=1, n, (-1)^k*k^2) - Benoit Cloitre, Aug 29 2002
a(n) = ((n+2)/n)*a(n-1)
Sum(n=1..infinity, 1/a(n)) = 2. - Jon Perry, Jul 13 2003
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, Jul 18 2003
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, Aug 19 2003
a(n+1) is the determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+1, i) - Benoit Cloitre, Aug 19 2003
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
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, Nov 04 2003
a(n)+a(n+1) = (n+1)^2. - N. J. A. Sloane, Feb 19 2004
a(n) = a(n-2)+2n-1. - Paul Barry, Jul 17 2004
a(n) = Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk, Oct 24 2004
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, Oct 26 2004
a(0) = 0, a(1) = 1, a(n) = 2*a(n-1)-a(n-2)+1 - Miklos Kristof, Mar 09 2005
a(n) = Sum_{k = 1...n} phi(k)*floor(n/k) = Sum{k = 1...n} A000010(k)*A010766(n, k) (R. Dedekind). - Vladeta Jovovic, Feb 05 2004
a(n) = A108299(n+3,4) = -A108299(n+4,5). - Reinhard Zumkeller, Jun 01 2005
a(n) = floor((2n+1)^2/8) - Paul Barry, May 29 2006
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, Jul 29 2006
(a(n))^2+(a(n+1))^2 = a((n+1)^2) [R B Nelsen, Math Mag 70 (2) (1997) p 130]. - R. J. Mathar, Nov 22 2006
a(n)*a(n+k)+a(n+1)*a(n+1+k) = a((n+1)*(n+1+k)). Generalizes previous formula dated Nov 22 2006 [and the third comment by J. M. Bergot dated May 22 2012]. Charlie Marion, Feb 04 2011
a(n) = A023896(n) + A067392(n). - Lekraj Beedassy, Mar 02 2007
Sum_{k, 0<=k<=n}a(k)*A039599(n,k)=A002457(n-1), for n>=1 . - Philippe DELEHAM, Jun 10 2007
A general formula for polygonal numbers is P(k,n) = (k-2)(n-1)n/2 + n = n + (k-2)*A000217(n-1), for n >= 1, k >= 3. - Omar E. Pol, Apr 28 2008 and Mar 31 2013
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 Janjic, Dec 20 2008]
a(n) = A000124(n-1) + (n-1) for n >= 2. a(n) = A000124(n) - 1. A000124(n) = central polygonal numbers. [From Jaroslav Krizek, Jun 16 2009]
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, Jun 28 2009]
a(n) = A034856(n+1) - A005408(n) = A005843(n) + A000124(n) - A005408(n) = A000124(n) - 1. [From Jaroslav Krizek, Sep 05 2009]
With offset 1, a(n)=1/2*floor(n^3/(n+1)) [From Gary Detlefs, Feb 14 2010]
a(n) = 4*a(floor(n/2))+(-1)^(n+1)*floor((n+1)/2). [Bruno Berselli, May 23 2010]
a(n) = 3*a(n-1)-3*a(n-2)+a*(n-3); a(0)=0,a(1)=1. [From M. Dols (markdols99(AT)yahoo.com), Aug 20 2010]
a(n) = sqrt((sum(i^3,{i,1,n}))) [Zak Seidov, Dec 07 2010]
For n>0 a(n)=1/(Integral_{x=0..Pi/2} 4*(sin(x))^(2*n-1)*(cos(x))^3). [From Francesco Daddi, Aug 02 2011]
a(n) = A110654(n) * A008619(n). [Reinhard Zumkeller, Aug 24 2011]
a(2k-1) = A000384(k), a(2k) = A014105(k), k>0. - Omar E. Pol, Sep 13 2011
a(n) = A026741(n) * A026741(n+1). [Charles R Greathouse IV, Apr 01, 2012]
a(n) + a(a(n)) + 1 = a(a(n)+1). - J. M. Bergot, Apr 27 2012
a(n) = -s(n+1,n), where s(n,k) are the Stirling numbers of the first kind, A048994. [From Mircea Merca, May 03 2012]
a(n)*a(n+1) = a(sum(m=1..n, A005408(m)))/2, for n>=1. For example, if n=8, then a(8)*a(9) = a(80)/2 = 1620. [Ivan N. Ianakiev, May 27 2012]
a(n) = A002378(n)/2. - Omar E. Pol, Jun 07 2012
G.f.: x * (1 + 3x + 6x^2 + ...) = x * Product_{j>=0} (1+x^(2^j))^3 = x * A(x) * A(x^2) * A(x^4) * ... , where A(x) = (1 + 3x + 3x^2 + x^3). - Gary W. Adamson, Jun 26 2012
E.g.f.: x*U(0) where U(k)= 1 + x/(2 - 4/(2 + (k+1)/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 18 2012
G.f.: G(0) where G(k)= 1 + (2*k+3)*x/(2*k+1 - x*(k+2)*(2*k+1)/(x*(k+2) + (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
a(n) = (A001318(n) + A085787(n))/2. - Omar E. Pol, Jan 11 2013
a(n) = A002088(n) + A063985(n). - Reinhard Zumkeller, Jan 21 2013
G.f.: x + 3*x^2/(Q(0)-3*x) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) + a(n+1) + a(n+2) + a(n+3) + n = a(2n+4). - Ivan N. Ianakiev, Mar 16 2013
a(n) + a(n+1) + ... + a(n+8) + 6n = a(3n+15). - Charlie Marion, Mar 18 2013
a(n) + a(n+1) + ... + a(n+20) + 2n^2 + 57n = a(5n+55). - Charlie Marion, Mar 18 2013
3*a(n) + a(n-1) = a(2n), for n>0. - Ivan N. Ianakiev, Apr 05 2013
a(n+1) = det(C(i+2,j+1), 1 <= i,j <= n), where C(n,k) are binomial coefficients. [Mircea Merca, Apr 06 2013]
a(n)^2 = sum(i=1..n, sum(j=1..n, ij)), for n>0. - Ivan N. Ianakiev, May 01 2013
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EXAMPLE
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x + 3*x^2 + 6*x^3 + 10*x^4 + 15*x^5 + 21*x^6 + 28*x^7 + 36*x^8 + 45*x^9 + ...
When n=3, a(3) = 4*3/2 = 6.
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
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MAPLE
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A000217 := proc(n) n*(n+1)/2; end;
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)
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
A000217:=-1/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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Table[(m^2 - m)/2, {m, 54}] (* Zerinvary Lajos, Mar 24 2007 *)
Array[ #*(# - 1)/2 &, 54] (* From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009 *)
FoldList[#1 + #2 &, 0, Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[Range[0, 70]] (* Harvey P. Dale, Sep 09 2012 *)
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PROG
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(PARI) A000217(n) = {n * (n + 1) / 2}
(PARI) is_A000217(n)=n*2==(1+n=sqrtint(2*n))*n \\ M. F. Hasler, May 24 2012
(Haskell)
a000217 n = a000217_list !! n
a000217_list = scanl1 (+) [0..] -- Reinhard Zumkeller, Sep 23 2011
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CROSSREFS
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Cf. A007318, A002024, A000124, A002378, A000292, A000330, A000096, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A051942, A101859, A001477, A046092, A001082, A036666, A062717, A028347, A087475, A006011, A002415, A010054, A210569.
A diagonal of A008291. a(n) = A110555(n+2, 2).
a(n) = A110449(n, 0).
Cf. A143320 [From Gary W. Adamson, Aug 07 2008]
Cf. A000396, A000668. [From Omar E. Pol, Sep 05 2008]
a(3*n)=A081266, a(4*n)=A033585, a(5*n)=A144312, a(6*n)=A144314. [From Reinhard Zumkeller, Sep 17 2008]
Cf. A008953, A008954. [From Reinhard Zumkeller, May 12 2010]
n-gonal numbers: A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.
Column 2 of A195152.
Numbers of the form n*t(n+k,h)-(n+k)*t(n,h), where t(i,h) = i*(i+2*h+1)/2 for any h (for A000217 is k=1): A005563, A067728, A140091, A140681, A212331.
Sequence in context: A105339 A089594 A161680 * A105340 A176659 A109811
Adjacent sequences: A000214 A000215 A000216 * A000218 A000219 A000220
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KEYWORD
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nonn,core,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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