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A109526 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;1,0,0,1]. 0
0, 2, 2, 4, 2, 4, 5, 8, 8, 12, 16, 23, 29, 41, 56, 78, 105, 146, 201, 278, 381, 527, 727, 1004, 1383, 1910, 2636, 3639, 5020, 6930, 9565, 13203, 18221, 25151, 34715, 47917, 66136, 91287, 126001, 173917, 240051, 331338, 457338, 631254, 871303, 1202641 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..45.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-1,0,0,-1).

FORMULA

G.f.: (-3*x^6 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 + 2)/(x^8 + x^5 - 2*x^4 - x + 1).

EXAMPLE

a(7)=8 because P^7=[0,0,0,1;1,0,0,0;0,1,0,0;0,0,1,0], T^7=[1,1,1,2;2,1,1,3;3,2,1,4;4,3,2,5] and so P^7+T^7=[1,1,1,3;3,1,1,3;3,3,1,4;4,3,3,5] and now a(7)=1+1+2*3=8.

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, 0, 0, 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..50);

MATHEMATICA

v[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, 0, 0, 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

Cf. A000045, A000213, A000931.

Sequence in context: A330882 A205138 A233763 * A309894 A260723 A059214

Adjacent sequences:  A109523 A109524 A109525 * A109527 A109528 A109529

KEYWORD

nonn,easy

AUTHOR

Roger L. Bagula, Jun 17 2005

STATUS

approved

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Last modified July 4 18:38 EDT 2020. Contains 335448 sequences. (Running on oeis4.)