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A350473 a(n) = Fibonacci(n+1)^3 - Fibonacci(n-1)^3. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
See A346513 for Fibonacci(n+1)^3 - Fibonacci(n)^3.
LINKS
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.
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
Sequence in context: A026617 A240261 A232605 * A349309 A262110 A319425
KEYWORD
nonn,easy
AUTHOR
Greg Dresden, Jan 01 2022
STATUS
approved

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Last modified June 17 06:25 EDT 2024. Contains 373432 sequences. (Running on oeis4.)