A109523 a(n) is 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; 1,1,1].

0, 2, 2, 2, 5, 8, 13, 25, 45, 81, 150, 275, 504, 928, 1706, 3136, 5769, 10610, 19513, 35891, 66013, 121415, 223318, 410745, 755476, 1389538, 2555758, 4700770, 8646065, 15902592, 29249425, 53798081, 98950097, 181997601, 334745778

Offset

0,2

Links

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

Example

a(7)=25 because
P^7 = [0,1,0; 0,0,1; 1,0,0],
T^7 = [7,11,13; 13,20,24; 24,37,44] and so
P^7 + T^7 = [7,12,13; 13,20,25; 25,37,44].

Maple

with(linalg): a:=proc(n) local P, T, v, 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, 1]): v:=matrix(3, 1, [0, 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] end: 0, seq(a(n), n=1..40);

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, 1}}; 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.

Sequence in context: A322176 A039886 A367087 * A339613 A008295 A216694

Adjacent sequences: A109520 A109521 A109522 * A109524 A109525 A109526

Keyword

nonn

Author

Roger L. Bagula, Jun 17 2005

Status

approved