6,2

Number of triangles (all of whose vertices lie inside the circle) formed when n points in general position on a circle are joined by straight lines - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 25, 2000.

Figurate numbers based on 6-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 13 of these numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 28 2004

a(n) = A110555(n+1,6). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

a(n) is the number of terms in the expansion of (a_1+a_2+a_3+a_4+a_5+a_6+a_7)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007

Product of six consecutive numbers divided by 6! - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Only prime in this sequence is 7 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

With a different offset, number of n-permutations (n>=6) of 2 objects: u,v, with repetition allowed, containing exactly six (6) u's. Example: a(1)=7 because we have uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu and vuuuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008

6-dimensional triangular numbers, sixth partial sums of binomial transform of [1,0,0,0,...]. a(n+6)=sum{i=0,n,C(n+6,i+6)*b(i)}, where b(i)=[1,0,0,0,...], a(n+6)=C(n+6,6). [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009, R. J. Mathar, Jul 07 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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.

Leo Moser, Mathematics Magazine, 26 (March, 1953), p. 226.

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

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

Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 11, #32

T. D. Noe, Table of n, a(n) for n=6..1000

Index entries for sequences related to linear recurrences with constant coefficients

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, 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.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 256

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

H. K. Kim, On Regular Polytope Numbers, Journal: Proc. Amer.Math. Soc. 131 (2003), 65-75, as PDF file.

J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

G.f.: x^6/(1-x)^7.

(x^6-15*x^5+85*x^4-225*x^3+274*x^2-120*x)/720

Conjecture: a(n+3) = Sum{0<=k, l, m<=n; k+l+m<=n} k*l*m. - Ralf Stephan (ralf(AT)ark.in-berlin.de), May 06 2005

Convolution of the nonnegative numbers (A001477) with the hexagonal numbers (A000389). Also convolution of the triangular numbers (A000217) with the tetrahedral numbers (A000292) - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007

a(n)=numbperm (n,6)/720, n>=6 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

a(n)=n(n-1)(n-2)(n-3)(n-4)(n-5)/720 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007, R. J. Mathar, Jul 07 2009

Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, 0, 0, 0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2008]

a(4) = 84 = (1, 3, 3, 1) dot (1, 6, 15, 20) = (1 + 18 + 45 + 20). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 02 2008]

A000579 := n->binomial(n, 6);

ZL := [S, {S=Prod(B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=7..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

seq(numbperm (n, 6)/720, n=6..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

A000579:=-1/(z-1)**7; [S. Plouffe in his 1992 dissertation, referring to offset 0.]

seq(binomial(n+6, 6)*1^n, n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008

restart: G(x):=x^6*exp(x): f[0]:=G(x): for n from 1 to 39 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/6!, n=6..39); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

Table[Binomial[n, 6], {n, 6, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006

Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)/720, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Cf. A053135, A053128, A000580, A000581, A000582.

Cf. A000217, A000292, A000332, A000389.

Adjacent sequences: A000576 A000577 A000578 this_sequence A000580 A000581 A000582

Sequence in context: A049018 A008489 A023032 this_sequence A049017 A019501 A145456

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006

Some formulas that referred to other offsets corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009