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A006522
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4-dimensional analog of centered polygonal numbers. Also number of regions created by sides and diagonals of a convex n-gon in general position.
(Formerly M3413)
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19
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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
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OFFSET
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0,5
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COMMENTS
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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
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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
Sum_{n>=3} 1/a(n) = 66/25 - (4/5)*sqrt(3/13)*Pi*tanh(sqrt(39)*Pi/2). - Amiram Eldar, Aug 23 2022
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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):
A006522:=-(1-z+z**2)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence except for three leading terms
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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 (* 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}]];
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PROG
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(Magma) [Binomial(n, 4)+Binomial(n-1, 2): n in [0..40]]; // Vincenzo Librandi, Jun 09 2013
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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