0,6

Number of intersection points of diagonals of convex n-gon.

Also the number of equilateral triangles with vertices in a equilateral triangular array of points with n rows (offset 1), with any orientation. - Ignacio Larrosa Canestro (ignacio.larrosa(AT)eresmas.net), Apr 09 2002

Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - rgwv, Aug 02 2002

For n>0 a(n)=(-1/8)*coefficient of x in Zagier's polynomial P_(2n,n) (Zagier's polynomials are used by pari-gp for acceleration of alternating or positive series)

Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n+1)*(n+2)*(n+3))/4!) - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004

a(n) = A110555(n+1,4). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005

Maximal number of crossings that can be created by connecting n vertices with straight lines. - Cameron Redsell-Montgomerie (credsell(AT)uoguelph.ca), Jan 30 2007

If X is an n-set and Y a fixed (n-1)-subset of X then a(n) is equal to the number of 4-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Aug 15 2007

Product of four consecutive numbers divided by 24 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Only prime in this sequence is 5 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

For strings consisting entirely of 0s and 1s, the number of unique arrangements of four 1s such that 1s are not adjacent. The shortest possible string is 7 characters, of which there is only one solution: 1010101, corresponding to a(5). An eight character string has 5 solutions, nine has 15, ten has 35, and so on, congruent to A000332. - Gil Broussard (gil_broussard(AT)bellsouth.net), Mar 19 2008

Apart from the first 4 zeros, this sequence also represents the partial sums of oblong numbers. That is, a(n)=n(n+1)(n+2)(n+3)/24 (n>=1). - Mohammad K. Azarian (azarian(AT)evansville.edu), Apr 03 2008

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.

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.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.

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. 7.

Norbert Kaufman and R. H. Koch, Amer. Math. Monthly, 54 (Jun, 1947), p. 344.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 53, #191

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1002

Milan Janjic, Two Enumerative Functions

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

X. Acloque, Polynexus Numbers.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Th. Gruner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 254

Hyun Kwang Kim, On Regular Polytope Numbers

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

Les Reid, Counting Triangles in an Array

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Pentatope

n*(n+1)*(n+2)*(n+3)/24. G.f. if offset -1: 1/(1-x)^5.

a(n)=(n+4)/n*a(n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 26 2003

a(n)=sum(k=1, n-3, sum(i=1, k, i*(i+1)/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 15 2003

Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...} - Jon Perry (perry(AT)globalnet.co.uk), Jun 25 2003

a(n) = ([(n^5-(n-1)^5)-(n^3-(n-1)^3)]/24) - ((n^5-(n-1)^5-1)/30); a(n) = A006322 - A006325. - Xavier Acloque Oct 20 2003

G.f.: x^4/(1-x)^5 - Jon Perry (perry(AT)globalnet.co.uk), Mar 31 2004

a(4n+2) = Pyr(n+4, 4n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = [(A-2)*B^3 + 3*B^2 - (A-5)*B]/6; For all positive integers i, and the pentagonal number function P(x) = x*(3*x-1)/2: a(3i-2) = P(P(i)) and a(3i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 15 2004

First differences of A000389(n) - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 19 2004

The sum of the first n tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005

a(n)=(n-4)(n-3)(n-2)(n-1)/24 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Starting (1, 5, 15, 35,...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2007

A000332 := n->binomial(n, 4); [seq(binomial(n, 4), n=3..100)];

ZL := [S, {S=Prod(B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=1..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

with(combinat):seq(numbcomp(i+4, i), i=-3..40) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007

A000332:=-1/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]

seq(sum(binomial(n, k+1), k=3..3), n=0..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007

Table[ Binomial[n, 4], {n, 3, 45} ]

Table[Sqrt[StirlingS2[i+1, i]*(-StirlingS1[i+3, i])/6] , {i, 0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

Table[(n - 4)(n - 3)(n - 2)(n - 1)/24, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Cf. A053134, A053126, A000389, A000579-A000582, A075733, A006322, A006325.

Cf. also A000583, A014820, A092181, A092182, A092183.

Partial sums of A001044.

Cf. A000217, A000292.

Adjacent sequences: A000329 A000330 A000331 this_sequence A000333 A000334 A000335

Sequence in context: A008487 A000743 A090580 this_sequence A049016 A137360 A100355

nonn,easy,nice

njas

More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000