OFFSET
0,2
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,1,1,2,-1,-1,-1,-1).
EXAMPLE
a(7)=31 because P^7=[0,0,0,1;1,0,0,0;0,1,0,0;0,0,1,0], T^7=[4,6,7,8;8,12,14,15;15,23,27,29;29,44,52,56] and so P^7+T^7=[4,6,7,9;9,12,14,15;15,24,27,29;29,44,53,56]; so a(7)=6+7+2*9=31.
MAPLE
with(linalg): a:=proc(n) local P, T, k: P[1]:=matrix(4, 4, [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0]): T[1]:=matrix(4, 4, [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1]):for k from 2 to n do P[k]:=multiply(P[1], P[k-1]): T[k]:=multiply(T[1], T[k-1]) od: evalm(P[n]+T[n])[1, 2]+evalm(P[n]+T[n])[1, 3]+2*evalm(P[n]+T[n])[1, 4] end: 0, seq(a(n), n=1..40);
MATHEMATICA
[0] = {0, 1, 1, 2}; w[0] = {0, 1, 1, 2}; M4 = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}}; Mt = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 1, 1}}; v[n_] := v[n] = M4.v[n - 1] w[n_] := w[n] = Mt.w[n - 1] a = Table[(w[n] + v[n])[[1]], {n, 0, 50}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Jun 17 2005
STATUS
approved