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A005732
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a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).
(Formerly M4514)
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4
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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
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OFFSET
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3,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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Claudi Alsina and Roger B. Nelson, A Panoply of Polygons, Dolciani Math. Expeditions, AMS/MAA (2023) Vol. 58, see page 7.
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FORMULA
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G.f.: x^3*(-1-x+x^3) / (x-1)^7 . - Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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Table[Binomial[n+3, 6]+Binomial[n+1, 5]+Binomial[n, 5], {n, 3, 40}] (* Harvey P. Dale, Apr 09 2011 *)
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PROG
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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
(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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