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a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).
(Formerly M4514)
4

%I M4514 #58 Feb 23 2023 01:29:34

%S 1,8,35,111,287,644,1302,2430,4257,7084,11297,17381,25935,37688,53516,

%T 74460,101745,136800,181279,237083,306383,391644,495650,621530,772785,

%U 953316,1167453,1419985,1716191,2061872,2463384,2927672,3462305,4075512,4776219,5574087,6479551

%N a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).

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

%C Equals binomial transform of [1, 7, 20, 29, 22, 8, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Jun 13 2008

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

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

%H T. D. Noe, <a href="/A005732/b005732.txt">Table of n, a(n) for n = 3..1000</a>

%H Claudi Alsina and Roger B. Nelson, <a href="https://bookstore.ams.org/view?ProductCode=DOL/58">A Panoply of Polygons</a>, Dolciani Math. Expeditions, AMS/MAA (2023) Vol. 58, see page 7.

%H R. J. Cormier and R. B. Eggleton, <a href="http://www.jstor.org/stable/2690116">Counting by correspondence</a>, Math. Mag., 49 (1976), 181-186.

%H R. K. Guy & M. E. Larsen, <a href="/A005732/a005732.pdf">Correspondence, 1986-87</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/triangle_counting">Number of triangles for a convex n-gon</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f.: x^3*(-1-x+x^3) / (x-1)^7 . - _Simon Plouffe_ in his 1992 dissertation

%F a(2n-1) = A006600(2n-1) for n > 1; a(2n) = A006600(2n) + A260417(n) for n > 1. - _Jonathan Sondow_, Jul 25 2015

%t Table[Binomial[n+3,6]+Binomial[n+1,5]+Binomial[n,5],{n,3,40}] (* _Harvey P. Dale_, Apr 09 2011 *)

%o (Magma) [Binomial(n+3, 6) + Binomial(n+1, 5) +Binomial(n,5): n in [3..100]]; // _Vincenzo Librandi_, Apr 10 2011

%o (Haskell)

%o a005732 n = a005732_list !! (n-3)

%o a005732_list = 1 : 8 : f (drop 5 a007318_tabl) where

%o f (us:pss@(vs:_:ws:_)) = (us !! 5 + vs !! 5 + ws !! 6) : f pss

%o -- _Reinhard Zumkeller_, Mar 11 2014

%o (PARI) a(n)=binomial(n+3,6) + binomial(n+1,5) + binomial(n,5) \\ _Charles R Greathouse IV_, Feb 19 2017

%Y Often confused with A006600.

%Y Cf. A007318, A260417.

%K nonn,easy,nice

%O 3,2

%A _N. J. A. Sloane_

%E Thanks to _Joshua Zucker_, Ted Alper and Joe Keane for clarifying the connection with A006600.