A093043 a(n) = Jacobsthal(n)*Fibonacci(n-1).

0, 0, 1, 3, 10, 33, 105, 344, 1105, 3591, 11594, 37565, 121485, 393264, 1272413, 4117971, 13325450, 43123017, 139547457, 451587592, 1461364025, 4729080015, 15303613546, 49523551333, 160261550085, 518617316448, 1678280815525, 5431030925499, 17575185066730, 56874493910481

Offset

0,4

Comments

Form a graph from a triangle and its midpoint triangle. a(n) counts walks of length n between two vertices of the original triangle.

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

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

Formula

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

a(n) = A001045(n)*A000045(n-1).

a(n) = (2^n/3-(-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, Jun 05 2026

Mathematica

A093043[n_] := Round[2^n/3]*Fibonacci[n-1]; Array[A093043, 30, 0] (* Paolo Xausa, Jun 06 2026 *)
(* Alternative: *)
LinearRecurrence[{1, 7, 2, -4}, {0, 0, 1, 3}, 30] (* Paolo Xausa, Jun 06 2026 *)

Crossrefs

Cf. A000045, A001045.

Sequence in context: A115240 A027989 A096483 * A061566 A082398 A053156

Adjacent sequences: A093040 A093041 A093042 * A093044 A093045 A093046

Keyword

easy,nonn,changed

Author

Paul Barry, Mar 22 2004

Extensions

More terms from Paolo Xausa, Jun 06 2026

Status

approved