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A006522 4-dimensional analogue of centered polygonal numbers. Also number of regions created by sides and diagonals of n-gon in general position.
(Formerly M3413)
15
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; text; internal format)
OFFSET

0,5

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 Janjic, 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)

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Math Forum, Regions of a circle Cut by Chords to n points.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Polygon Diagonal.

FORMULA

a(n)=binomial(n, 4)+ binomial(n-1, 2)

binomial(n,2)+binomial(n,3)+binomial(n,4), n>=-1. - Zerinvary Lajos, 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). [Harvey P. Dale, Jul 11 2011]

G.f.: -(((x-1)*x*(x*(4*x-5)+5)+1)/(x-1)^5). [Harvey P. Dale, Jul 11 2011]

a(n) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 24)/24. - T. D. Noe, Oct 16 2013

EXAMPLE

For a pentagon in general position, 11 regions are formed (Comtet, Fig. 20, p. 74).

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, Jul 23 2006

A006522:=-(1-z+z**2)/(z-1)**5; [Simon 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, Dec 14 2007

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 Joseph Stephan Orlovsky, 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] (* Harvey P. Dale, Jul 11 2011 *)

CoefficientList[Series[-(((x - 1) x (x (4 x - 5) + 5) + 1) / (x - 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

PROG

(MAGMA) [Binomial(n, 4)+Binomial(n-1, 2): n in [0..40]]; // Vincenzo Librandi, Jun 09 2013

CROSSREFS

Partial sums of A004006.

Sequence in context: A110610 A051462 A006004 * A036837 A215052 A011851

Adjacent sequences:  A006519 A006520 A006521 * A006523 A006524 A006525

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 30 17:45 EDT 2014. Contains 247475 sequences.