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%I #10 Mar 12 2014 16:37:00
%S 1,3,5,-9,-99,-297,389,8655,46573,122823,-120491,-3003393,-15885315,
%T -40246281,50400229,1040606127,5296630541,12512952615,-22872751243,
%U -368600380833,-1789336379619,-3926384911017,9502037022725,129579396089871,602116408170541,1219711972804743
%N 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]).
%F 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.
%F 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
%p 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);
%t 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}]
%Y Cf. A000952, A074872, A085340.
%K sign
%O 1,2
%A _Roger L. Bagula_, Oct 05 2006
%E Edited by _N. J. A. Sloane_, Oct 15 2006
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