A005732 a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5). (Formerly M4514)

1, 8, 35, 111, 287, 644, 1302, 2430, 4257, 7084, 11297, 17381, 25935, 37688, 53516, 74460, 101745, 136800, 181279, 237083, 306383, 391644, 495650, 621530, 772785, 953316, 1167453, 1419985, 1716191, 2061872, 2463384, 2927672, 3462305, 4075512, 4776219, 5574087, 6479551

Offset

3,2

Comments

Place n points in general position on a circle, join them in all possible ways; how many triangles can be seen?

Equals binomial transform of [1, 7, 20, 29, 22, 8, 1, 0, 0, 0, ...]. - Gary W. Adamson, Jun 13 2008

References

C. L. Liu, Introduction to Combinatorial Analysis. McGraw-Hill, NY, 1968, p. 20.

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

Links

T. D. Noe, Table of n, a(n) for n = 3..1000

Claudi Alsina and Roger B. Nelson, A Panoply of Polygons, Dolciani Math. Expeditions, AMS/MAA (2023) Vol. 58, see page 7.

R. J. Cormier and R. B. Eggleton, Counting by correspondence, Math. Mag., 49 (1976), 181-186.

R. K. Guy & M. E. Larsen, Correspondence, 1986-87

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

T. Sillke, Number of triangles for a convex n-gon

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

G.f.: x^3*(-1-x+x^3) / (x-1)^7 . - Simon Plouffe in his 1992 dissertation

a(2n-1) = A006600(2n-1) for n > 1; a(2n) = A006600(2n) + A260417(n) for n > 1. - Jonathan Sondow, Jul 25 2015

Mathematica

Table[Binomial[n+3, 6]+Binomial[n+1, 5]+Binomial[n, 5], {n, 3, 40}]  (* Harvey P. Dale, Apr 09 2011 *)

Prog

(Magma) [Binomial(n+3, 6) + Binomial(n+1, 5) +Binomial(n, 5): n in [3..100]]; // Vincenzo Librandi, Apr 10 2011
(Haskell)
a005732 n = a005732_list !! (n-3)
a005732_list = 1 : 8 : f (drop 5 a007318_tabl) where
   f (us:pss@(vs:_:ws:_)) = (us !! 5 + vs !! 5 + ws !! 6) : f pss
-- Reinhard Zumkeller, Mar 11 2014
(PARI) a(n)=binomial(n+3, 6) + binomial(n+1, 5) + binomial(n, 5) \\ Charles R Greathouse IV, Feb 19 2017

Crossrefs

Often confused with A006600.

Cf. A007318, A260417.

Sequence in context: A244882 A006600 A161456 * A162211 A161717 A162494

Adjacent sequences: A005729 A005730 A005731 * A005733 A005734 A005735

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Thanks to Joshua Zucker, Ted Alper and Joe Keane for clarifying the connection with A006600.

Status

approved