A307082 Expansion of 1/(1 - x/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 5*x/(1 - 8*x/(1 - ... - Fibonacci(k)*x/(1 - ...)))))))), a continued fraction.

1, 1, 2, 6, 26, 164, 1540, 22068, 492616, 17378968, 977896328, 88256247312, 12819022165520, 3002745820555664, 1135759674922075168, 694219521332053782624, 686053892556368634929824, 1096476587053610841771551296, 2834651494015025836540377942080

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..99

Formula

a(n) ~ c * phi^(n*(n+1)/2) / 5^(n/2), where phi = A001622 is the golden ratio and c = 10.15498753508843821457456033641336796744756370048241257586748102558791... - Vaclav Kotesovec, Sep 18 2021

Maple

b:= proc(x, y) option remember; (F->
     `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-x+1), 0)+
     `if`(y>x, b(x, y-1), 0)))(combinat[fibonacci])
    end:
a:= n-> b(n$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Nov 12 2023

Mathematica

nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-Fibonacci[k] x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A000045, A001622, A206741, A307083.

Sequence in context: A180349 A141713 A005272 * A178089 A363003 A002449

Adjacent sequences: A307079 A307080 A307081 * A307083 A307084 A307085

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 23 2019

Status

approved