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A261031 Euler transform of Lucas numbers.

%I A261031 #15 Nov 11 2020 07:10:53
%S A261031 1,1,4,8,21,44,103,217,477,999,2116,4373,9055,18464,37576,75725,
%T A261031 152047,303158,602085,1189242,2340065,4584027,8947865,17399906,
%U A261031 33725509,65153150,125493914,241011287,461611911,881806114,1680336592,3194346093,6058770147,11466709780
%N A261031 Euler transform of Lucas numbers.
%H A261031 Vaclav Kotesovec, <a href="/A261031/b261031.txt">Table of n, a(n) for n = 0..2000</a>
%F A261031 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
%F A261031 G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - _Ilya Gutkovskiy_, May 30 2018
%p A261031 L:= proc(n) option remember; `if`(n<2, 2-n, L(n-2)+L(n-1)) end:
%p A261031 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A261031       L(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A261031     end:
%p A261031 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 12 2017
%t A261031 CoefficientList[Series[Product[1/(1 - x^k)^LucasL[k], {k, 1, 30}], {x, 0, 30}], x]
%o A261031 (SageMath) # uses[EulerTransform from A166861]
%o A261031 a = BinaryRecurrenceSequence(1, 1, 2)
%o A261031 b = EulerTransform(a)
%o A261031 print([b(n) for n in range(34)]) # _Peter Luschny_, Nov 11 2020
%Y A261031 Cf. A000032, A166861.
%K A261031 nonn
%O A261031 0,3
%A A261031 _Vaclav Kotesovec_, Aug 07 2015