A109525 a(n)=the sum of the (1,2)- and (1,3)-entries and twice the (1,4)-entry of the matrix P^n + T^n, where the 4 X 4 matrices P and T are defined by P=[0,1,0,0;0,0,1,0;0,0,0,1;1,0,0,0] and T=[0,1,0,0;0,0,1,0;0,0,0,1;1,1,1,1].

0, 2, 2, 4, 4, 9, 16, 31, 56, 109, 209, 403, 773, 1491, 2873, 5538, 10671, 20570, 39649, 76426, 147312, 283954, 547338, 1055028, 2033628, 3919945, 7555936, 14564535, 28074040, 54114453, 104308961, 201061987, 387559437, 747044835, 1439975217

Offset

0,2

Links

Table of n, a(n) for n=0..34.

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

Sequence in context: A364752 A035054 A099537 * A243330 A318838 A231523

Adjacent sequences: A109522 A109523 A109524 * A109526 A109527 A109528

Keyword

nonn,easy

Author

Roger L. Bagula, Jun 17 2005

Status

approved