A123220 a(n)=the (1,1)-term of M^(n-1), where M=matrix(5,5, [3,-1,-1,-1,-1; 1,3,-1,-1,-1; 1,1,3,-1,-1; 1,1,1,3,-1; 1,1,1,1,3]).

1, 3, 5, -9, -99, -297, 389, 8655, 46573, 122823, -120491, -3003393, -15885315, -40246281, 50400229, 1040606127, 5296630541, 12512952615, -22872751243, -368600380833, -1789336379619, -3926384911017, 9502037022725, 129579396089871, 602116408170541, 1219711972804743

Offset

1,2

Links

Table of n, a(n) for n=1..26.

Formula

a(1) = 1; a(2) = 3; a(3) = 5; a(4) = -9; a(5) = -99; a(n) = 15a(n-1)-100a(n-2)+360a(n-3)-680a(n-4)+528a(n-5) for n>= 6. The minimal polynomial of M is x^5-15x^4+100x^3-360x^2+680x-528, the coefficients of which yield the coefficients of the recurrence relation.

O.g.f.: -x*(1-12*x+60*x^2-144*x^3+136*x^4)/((3*x-1)*(176*x^4-168*x^3+64*x^2-12*x+1)). - R. J. Mathar, Dec 05 2007

Maple

a[1]:=1:a[2]:=3:a[3]:=5:a[4]:=-9:a[5]:=-99: for n from 6 to 26 do a[n]:=15*a[n-1]-100*a[n-2]+360*a[n-3]-680*a[n-4]+528*a[n-5] od: seq(a[n], n=1..26); with(linalg): M[1]:=matrix(5, 5, [3, -1, -1, -1, -1, 1, 3, -1, -1, -1, 1, 1, 3, -1, -1, 1, 1, 1, 3, -1, 1, 1, 1, 1, 3]): for n from 2 to 25 do M[n]:=multiply(M[1], M[n-1]) od: 1, seq(M[n][1, 1], n=1..25);

Mathematica

M = {{3, -1, -1, -1, -1}, {1, 3, -1, -1, -1}, {1, 1, 3, -1, -1}, {1, 1, 1, 3, -1}, {1, 1, 1, 1, 3}}; w[1] = {1, 0, 0, 0, 0}; w[n_] := w[n] = M.w[n - 1]; a = Table[w[n][[1]], {n, 1, 30}]

Crossrefs

Cf. A000952, A074872, A085340.

Sequence in context: A336144 A215440 A163550 * A098339 A121622 A188983

Adjacent sequences: A123217 A123218 A123219 * A123221 A123222 A123223

Keyword

sign

Author

Roger L. Bagula, Oct 05 2006

Extensions

Edited by N. J. A. Sloane, Oct 15 2006

Status

approved