A005732 a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).
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
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).
Programs
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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
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Magma
[Binomial(n+3, 6) + Binomial(n+1, 5) +Binomial(n,5): n in [3..100]]; // Vincenzo Librandi, Apr 10 2011
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Mathematica
Table[Binomial[n+3,6]+Binomial[n+1,5]+Binomial[n,5],{n,3,40}] (* Harvey P. Dale, Apr 09 2011 *) -
PARI
a(n)=binomial(n+3,6) + binomial(n+1,5) + binomial(n,5) \\ Charles R Greathouse IV, Feb 19 2017
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
Extensions
Thanks to Joshua Zucker, Ted Alper and Joe Keane for clarifying the connection with A006600.
Comments