A006522 4-dimensional analog of centered polygonal numbers. Also number of regions created by sides and diagonals of a convex n-gon in general position.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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)). - Milan Janjic, Jan 24 2010

From Ant King, Sep 14 2011: (Start)

Consider the array formed by the polygonal numbers of increasing rank A139600

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)

Binomial transform of (1, -1, 1, 0, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Examples

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

References

• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
• Ross 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
• J. W. Freeman, The number of regions determined by a convex polygon, Math. Mag., 49 (1976), 23-25.
• Math Forum, Regions of a circle Cut by Chords to n points.
• V. Meally, Letter to N. J. A. Sloane, May 1975.
• Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Eric Weisstein's World of Mathematics, Polygon Diagonal.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Partial sums of A004006.

Cf. A000332, A000217.

Programs

Magma

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

Maple

A006522 := n->(1/24)*(n-1)*(n-2)*(n^2-3*n+12):
seq(A006522(n), n=0..40);
A006522:=-(1-z+z**2)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence except for three leading terms

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 (* 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 *)
a[n_] := If[n==0, 1, Sum[PolygonalNumber[n-k+1, k], {k, 0, n-2}]];
a /@ Range[0, 40] (* Jean-François Alcover, Jan 21 2020 *)

PARI

a(n)=1/24*n^4 - 1/4*n^3 + 23/24*n^2 - 7/4*n + 1 \\ Charles R Greathouse IV, Feb 09 2017

Formula

a(n) = binomial(n,4) + binomial(n-1,2) = A000332(n) + A000217(n-2).

a(n) = binomial(n-1,2) + binomial(n-1,3) + binomial(n-1,4). - Zerinvary Lajos, Jul 23 2006

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=4. - 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

For odd n, a(n) = A007678(n). - R. J. Mathar, Nov 22 2017

a(n) = a(3-n) for all n in Z. - Michael Somos, Nov 23 2021

Sum_{n>=3} 1/a(n) = 66/25 - (4/5)*sqrt(3/13)*Pi*tanh(sqrt(39)*Pi/2). - Amiram Eldar, Aug 23 2022