

A109523


a(n)=the sum of the (1,2) and (1,3)entries of the matrix P^n + T^n, where the 3 by 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].


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
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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[k1]): T[k]:=multiply(T[1], T[k1]) 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: A162145 A208050 A039886 * A008295 A216694 A116697
Adjacent sequences: A109520 A109521 A109522 * A109524 A109525 A109526


KEYWORD

nonn


AUTHOR

Roger L. Bagula, Jun 17 2005


STATUS

approved



