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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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12
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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; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n)=coeff(charpoly(A,x),x^(n-4)). [From Milan R. Janjic (agnus(AT)blic.net), Jan 24 2010]
From Ant King, Sep 14 2011: (Start)
Consider the array formed by the polygonal numbers of increasing rank
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, … A000217(n)
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, … A000290(n)
0, 1, 5, 12, 22, 35, 51, 70, 92, 117, … A000326(n)
0, 1, 6, 15, 28, 45, 66, 91, 120, 153, … A000384(n)
0, 1, 7, 18, 34, 55, 81, 112, 148, 189, … A000566(n)
0, 1, 8, 21, 40, 65, 96, 133, 176, 225, … A000567(n)
...
Then, for n>=2, a(n) is the diagonal sum of this polygonal grid.
(End)
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REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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
| 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
a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=4, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)- 5*a(n-4)+a(n-5) [From Harvey P. Dale, Jul 11 2011]
G.f.: -(((x-1)*x*(x*(4*x-5)+5)+1)/(x-1)^5) [From Harvey P. Dale, Jul 11 2011]
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EXAMPLE
| For a pentagon in general position, 11 regions are formed (Comtet, Fig. 20, p. 74).
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MAPLE
| 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
| 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]
Table[Binomial[n, 4]+Binomial[n-1, 2], {n, 0, 40}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {1, 0, 0, 1, 4}, 40] (* From Harvey P. Dale, Jul 11 2011 *)
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CROSSREFS
| Partial sums of A004006.
Sequence in context: A110610 A051462 A006004 * A036837 A011851 A193912
Adjacent sequences: A006519 A006520 A006521 * A006523 A006524 A006525
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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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