A093042 a(n) = Jacobsthal(n) * Fibonacci(n).

0, 1, 1, 6, 15, 55, 168, 559, 1785, 5814, 18755, 60787, 196560, 636323, 2058797, 6663030, 21561015, 69774527, 225792504, 730684103, 2364536625, 7651812246, 24761766811, 80130789371, 259308635040, 839140445275, 2715515402053, 8787592631574, 28437246796335, 92024864240359

Offset

0,4

Comments

Form a graph from a triangle and its midpoint triangle. This sequence counts walks of length n between a vertex of the original triangle and an adjacent vertex of the midpoint triangle.

Links

Table of n, a(n) for n=0..29.

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

Formula

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

a(n) = A001045(n) * A000045(n).

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

E.g.f.: (2 * (exp(-x/2) * sinh(sqrt(5)*x/2) + exp(x) * sinh(sqrt(5)*x))) / (3 * sqrt(5)). - Amiram Eldar, Jan 15 2026

Mathematica

LinearRecurrence[{1, 7, 2, -4}, {0, 1, 1, 6}, 30] (* Harvey P. Dale, Aug 17 2015 *)

Crossrefs

Cf. A000045, A001045.

Sequence in context: A119132 A073065 A190801 * A373569 A270624 A270675

Adjacent sequences: A093039 A093040 A093041 * A093043 A093044 A093045

Keyword

easy,nonn

Author

Paul Barry, Mar 22 2004

Status

approved