A261031 Euler transform of Lucas numbers.

1, 1, 4, 8, 21, 44, 103, 217, 477, 999, 2116, 4373, 9055, 18464, 37576, 75725, 152047, 303158, 602085, 1189242, 2340065, 4584027, 8947865, 17399906, 33725509, 65153150, 125493914, 241011287, 461611911, 881806114, 1680336592, 3194346093, 6058770147, 11466709780

Offset: 0

Author

Vaclav Kotesovec, Aug 07 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000032, A166861.

Programs

Maple

L:= proc(n) option remember; `if`(n<2, 2-n, L(n-2)+L(n-1)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      L(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 12 2017

Mathematica

CoefficientList[Series[Product[1/(1 - x^k)^LucasL[k], {k, 1, 30}], {x, 0, 30}], x]

SageMath

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(1, 1, 2)
b = EulerTransform(a)
print([b(n) for n in range(34)]) # Peter Luschny, Nov 11 2020

Formula

a(n) ~ phi^n / (2*sqrt(Pi)*n^(3/4)) * exp(-1 + 1/(2*sqrt(5)) + 2*sqrt(n) + s), where s = Sum_{k>=2} (2 + phi^k)/((phi^(2*k) - phi^k - 1)*k) = 0.9799662013576411396292209835034813778512885279062665867878344706... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 07 2015

G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018