A093044 A Jacobsthal Fibonacci product: a(n) = (2^n + 2*(-1)^n)*Fibonacci(n-1)/3.

1, 0, 2, 2, 12, 30, 110, 336, 1118, 3570, 11628, 37510, 121574, 393120, 1272646, 4117594, 13326060, 43122030, 139549054, 451585008, 1461368206, 4729073250, 15303624492, 49523533622, 160261578742, 518617270080, 1678280890550

Offset

0,3

Comments

Form a graph from a triangle and its midpoint triangle. This sequence counts closed walks of length n at a vertex of the original triangle.

Links

Table of n, a(n) for n=0..26.

Index entries for linear recurrences with constant coefficients, signature (1,7,2,-4).

Formula

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

a(n) = A078008(n)*A000045(n-1);

a(n) = (2^n/3+2*(-1)^n/3)*(((1+sqrt(5))/2)^(n-1)/sqrt(5)-((1-sqrt(5))/2)^(n-1)/sqrt(5)).

a(n) = a(n-1)+7*a(n-2)+2*a(n-3)-4*a(n-4). - Wesley Ivan Hurt, Apr 23 2021

Mathematica

LinearRecurrence[{1, 7, 2, -4}, {1, 0, 2, 2}, 30] (* Harvey P. Dale, Sep 01 2023 *)

Prog

(Magma) [(2^n + 2*(-1)^n)*Fibonacci(n-1)/3 : n in [0..30]]; // Wesley Ivan Hurt, Apr 23 2021

Crossrefs

Cf. A000045, A078008.

Sequence in context: A324919 A130306 A199127 * A151366 A184944 A033886

Adjacent sequences: A093041 A093042 A093043 * A093045 A093046 A093047

Keyword

easy,nonn

Author

Paul Barry, Mar 22 2004

Status

approved