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A135961 G.f.: A(x) = Sum_{n>=0} x^n/(1 - Fibonacci(n)*x).

%I A135961 #20 Sep 12 2015 11:00:19
%S A135961 1,1,2,3,5,10,25,79,318,1637,10753,89872,955537,12930173,222618066,
%T A135961 4874855543,135781292309,4811103270054,216847500834513,
%U A135961 12432143862756779,906625645142897790,84102571511631809865
%N A135961 G.f.: A(x) = Sum_{n>=0} x^n/(1 - Fibonacci(n)*x).
%H A135961 Vincenzo Librandi, <a href="/A135961/b135961.txt">Table of n, a(n) for n = 0..140</a>
%H A135961 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013
%F A135961 a(n) = Sum_{k=0..n} Fibonacci(k)^(n-k).
%F A135961 a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4) / 5^(n/4), where c = Sum_{k=-Infinity..Infinity} 5^(k/2)*((1+sqrt(5))/2)^(-k^2) = 3.5769727481316948565395...(see A219781) if n is even and c = Sum_{k=-Infinity..Infinity} 5^((k+1/2)/2)*((1+sqrt(5))/2)^(-(k+1/2)^2) = 3.5769727390073366345992... if n is odd. - _Vaclav Kotesovec_, Nov 29 2012
%e A135961 A(x) = 1 + x/(1-x) + x^2/(1-x) + x^3/(1-2x) + x^4/(1-3x) + x^5/(1-5x) +...
%t A135961 Flatten[{1, Table[Sum[Fibonacci[k]^(n-k), {k, 0, n}], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Nov 29 2012 *)
%o A135961 (PARI) a(n)=sum(k=0, n, fibonacci(k)^(n-k))
%Y A135961 Cf. A000045, A219781, A187780, A135741.
%K A135961 nonn
%O A135961 0,3
%A A135961 _Paul D. Hanna_, Dec 09 2007