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A005732
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C(n+3,6)+C(n+1,5)+C(n,5).
(Formerly M4514)
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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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 (qntmpkt(AT)yahoo.com), Jun 13 2008
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REFERENCES
| R. J. Cormier and R. B. Eggleton, Counting by correspondence, Math. Mag., 49 (1976), 181-186.
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
| T. D. Noe, Table of n, a(n) for n=3..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
T. Sillke, Number of triangles for a convex n-gon
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MAPLE
| A005732:=(-1-z+z**3)/(z-1)**7; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[Binomial[n+3, 6]+Binomial[n+1, 5]+Binomial[n, 5], {n, 3, 40}] (* From Harvey P. Dale, Apr 09 2011 *)
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PROG
| (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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CROSSREFS
| Often confused with A006600.
Sequence in context: A189592 A006600 A161456 * A162211 A161717 A162494
Adjacent sequences: A005729 A005730 A005731 * A005733 A005734 A005735
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Thanks to Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Ted Alper and Joe Keane for clarifying the connection with A006600.
More terms from Harvey P. Dale, Apr 09 2011.
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