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

0,3

Links

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

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

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]

Crossrefs

Cf. A000204, A001622, A206742, A307082.

Sequence in context: A091485 A201354 A112938 * A343709 A354147 A362475

Adjacent sequences: A307080 A307081 A307082 * A307084 A307085 A307086

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 23 2019

Status

approved