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

0,3

Links

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

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

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]

Prog

(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

Crossrefs

Cf. A000032, A166861.

Sequence in context: A061256 A180608 A244583 * A077921 A097076 A003608

Adjacent sequences: A261028 A261029 A261030 * A261032 A261033 A261034

Keyword

nonn

Author

Vaclav Kotesovec, Aug 07 2015

Status

approved