A135961 G.f.: A(x) = Sum_{n>=0} x^n/(1 - Fibonacci(n)*x).
1, 1, 2, 3, 5, 10, 25, 79, 318, 1637, 10753, 89872, 955537, 12930173, 222618066, 4874855543, 135781292309, 4811103270054, 216847500834513, 12432143862756779, 906625645142897790, 84102571511631809865
Offset: 0
Keywords
Examples
A(x) = 1 + x/(1-x) + x^2/(1-x) + x^3/(1-2x) + x^4/(1-3x) + x^5/(1-5x) +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..140
- V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013
Programs
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Mathematica
Flatten[{1, Table[Sum[Fibonacci[k]^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 29 2012 *) -
PARI
a(n)=sum(k=0, n, fibonacci(k)^(n-k))
Formula
a(n) = Sum_{k=0..n} Fibonacci(k)^(n-k).
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
Comments