A109524 a(n)=the sum of the (1,2)- and (1,3)-entries of the matrix P^n + T^n, where the 3 X 3 matrices P and T are defined by P=[0,1,0;0,0,1;1,0,0] and T=[0,1,0;0,0,1;1,1,0].

0, 2, 2, 1, 3, 3, 3, 5, 6, 7, 10, 13, 16, 22, 29, 37, 50, 66, 86, 115, 152, 200, 266, 352, 465, 617, 817, 1081, 1433, 1898, 2513, 3330, 4411, 5842, 7740, 10253, 13581, 17992, 23834, 31572, 41825, 55406, 73396, 97230, 128802, 170625, 226031, 299427, 396655

Offset

0,2

Links

Table of n, a(n) for n=0..48.

Example

a(7)=5 because P^7=[0,1,0;0,0,1;1,0,0], T^7=[1,2,2;2,3,2;2,4,3] and so P^7+T^7=[1,3,2;2,3,3;3,4,3].

Maple

with(linalg): a:=proc(n) local P, T, k: P[1]:=matrix(3, 3, [0, 1, 0, 0, 0, 1, 1, 0, 0]): T[1]:=matrix(3, 3, [0, 1, 0, 0, 0, 1, 1, 1, 0]):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] end: 0, seq(a(n), n=1..55);

Mathematica

v[0] = {0, 1, 1}; w[0] = {0, 1, 1}; M3 = {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}; Mt = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; v[n_] := v[n] = M3.v[n - 1] w[n_] := w[n] = Mt.w[n - 1] a = Table[(w[n] + v[n])[[1]], {n, 0, 50}]

Crossrefs

Cf. A000045, A000213, A000931.

Sequence in context: A112182 A112209 A240127 * A191521 A245370 A321341

Adjacent sequences: A109521 A109522 A109523 * A109525 A109526 A109527

Keyword

nonn

Author

Roger L. Bagula, Jun 17 2005

Status

approved