A350473 a(n) = Fibonacci(n+1)^3 - Fibonacci(n-1)^3.

0, 1, 7, 26, 117, 485, 2072, 8749, 37107, 157114, 665665, 2819609, 11944368, 50596649, 214331663, 907922170, 3846022173, 16292007901, 69014058568, 292348234421, 1238407008795, 5245976249306, 22222312038857, 94135224351601, 398763209531232, 1689188062337425

Offset

0,3

Comments

See A346513 for Fibonacci(n+1)^3 - Fibonacci(n)^3.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1595

Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Formula

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).

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

a(n) = (4/5)*Fibonacci(3*n) + (-1)^(n)*(3/5)*Fibonacci(n).

a(n) is the numerator of the continued fraction [1,...,1, 2 ,1,...,1, 2 ,1,1,...,1] with the first two runs of 1's of length n-2 and the last run of length n-1. For example, a(4)=117 which is the numerator of the continued fraction [1,1, 2 ,1,1, 2 ,1,1,1].

Mathematica

Differences[Fibonacci[Range[-1, 26]]^3, 1, 2]

Prog

(Python)
from sympy import fibonacci
def A350473(n): return fibonacci(n+1)**3-fibonacci(n-1)**3 # Chai Wah Wu, Jan 05 2022

Crossrefs

Cf. A000045, A110224, A346513.

Sequence in context: A026617 A240261 A232605 * A349309 A262110 A319425

Adjacent sequences: A350470 A350471 A350472 * A350474 A350475 A350476

Keyword

nonn,easy

Author

Greg Dresden, Jan 01 2022

Status

approved