A307083 - OEIS

%I #13 Nov 12 2023 12:55:15

%S 1,1,4,28,292,4408,97432,3231256,164789104,13170099856,1670220282544,

%T 338692348412320,110327835695333920,57892877044109184160,

%U 49019180876700301391680,67044425508546158335526080,148216012413625321252632612160,529829556146109541834263919553920

%N 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.

%H Alois P. Heinz, <a href="/A307083/b307083.txt">Table of n, a(n) for n = 0..97</a>

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

%p L:= proc(n) option remember; (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1] end:

%p b:= proc(x, y) option remember; `if`(x=0 and y=0, 1,

%p      `if`(x>0, b(x-1, y)*L(y-x+1), 0)+`if`(y>x, b(x, y-1), 0))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..17);  # _Alois P. Heinz_, Nov 12 2023

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

%Y Cf. A000204, A001622, A206742, A307082.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 23 2019