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A005732 a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).
(Formerly M4514)
3
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 (list; graph; refs; listen; history; text; internal format)
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

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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

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

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Last modified September 28 23:14 EDT 2016. Contains 276606 sequences.