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Search: id:A006522
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| A006522 |
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4-dimensional analogue of centered polygonal numbers. Also number of regions created by sides and diagonals of n-gon.
(Formerly M3413)
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+0
10
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| 1, 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, 375, 550, 781, 1079, 1456, 1925, 2500, 3196, 4029, 5016, 6175, 7525, 9086, 10879, 12926, 15250, 17875, 20826, 24129, 27811, 31900, 36425, 41416, 46904, 52921, 59500, 66675, 74481, 82954, 92131
(list; graph; listen)
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OFFSET
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0,5
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
J. W. Freeman, The number of regions determined by a convex polygon, Math. Mag., 49 (1976), 23-25.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 102.
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LINKS
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Math Forum, Regions of a circle Cut by Chords to n points.
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.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n)=binomial(n, 4)+ binomial(n-1, 2)
binomial(n,2)+binomial(n,3)+binomial(n,4), n>=-1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 23 2006
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EXAMPLE
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For a pentagon in general position, 11 regions are formed (Comtet, Fig. 20, p. 74).
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MAPLE
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A006522 := n->(1/24)*(n-1)*(n-2)*(n^2-3*n+12);
[seq(binomial(n, 2)+binomial(n, 3)+binomial(n, 4), n=-1..40)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 23 2006
A006522:=-(1-z+z**2)/(z-1)**5; [S. Plouffe in his 1992 dissertation. Gives sequence except for three leading terms.]
seq(sum(binomial(n, k+1), k=1..3), n=-1..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007
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MATHEMATICA
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a=2; b=3; s=4; lst={1, 0, 0, 1, s}; Do[a+=n; b+=a; s+=b; AppendTo[lst, s], {n, 2, 6!, 1}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 24 2009]
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CROSSREFS
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Partial sums of A004006.
Adjacent sequences: A006519 A006520 A006521 this_sequence A006523 A006524 A006525
Sequence in context: A110610 A051462 A006004 this_sequence A036837 A011851 A136395
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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