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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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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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