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

Author

Ilya Gutkovskiy, Mar 23 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A000045, A001622, A206741, A307083.

Programs

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]

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