A307083 Expansion of 1/(1 - x/(1 - 3*x/(1 - 4*x/(1 - 7*x/(1 - 11*x/(1 - 18*x/(1 - ... - Lucas(k)*x/(1 - ...)))))))), a continued fraction.

1, 1, 4, 28, 292, 4408, 97432, 3231256, 164789104, 13170099856, 1670220282544, 338692348412320, 110327835695333920, 57892877044109184160, 49019180876700301391680, 67044425508546158335526080, 148216012413625321252632612160, 529829556146109541834263919553920

Offset: 0

Author

Ilya Gutkovskiy, Mar 23 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..97

Crossrefs

Cf. A000204, A001622, A206742, A307082.

Programs

Maple

L:= proc(n) option remember; (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1] end:
b:= proc(x, y) option remember; `if`(x=0 and y=0, 1,
     `if`(x>0, b(x-1, y)*L(y-x+1), 0)+`if`(y>x, b(x, y-1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 12 2023

Mathematica

nmax = 17; CoefficientList[Series[1/(1 + ContinuedFractionK[-LucasL[k] x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * phi^(n*(n+1)/2), where phi = A001622 is the golden ratio and c = 5.62026823201787715079864730026619553810473701484813959397175006212578036... - Vaclav Kotesovec, Sep 18 2021