A102311 a(n) = Sum_{k=1..n} Fibonacci(k*(n-k)).

0, 1, 2, 7, 22, 86, 414, 2521, 19494, 191695, 2397716, 38148444, 772057396, 19875413009, 650843469738, 27110077916903, 1436411242814058, 96810095832996034, 8299583912379548210, 905077596297808256825, 125547805293905152853710, 22152679283963321048140511

Offset: 1

Author

Ralf Stephan, Jan 06 2005

Keywords

• nonn

Crossrefs

Cf. Antidiagonal sums of array A102310.

Cf. A000204, A001622.

Programs

Mathematica

Table[Sum[Fibonacci[k(n-k)],{k,n}],{n,30}] (* Harvey P. Dale, Jul 03 2019 *)

PARI

{a(n)=sum(k=1,n,fibonacci(k*(n-k)))}

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(m)*x^(m+1)/(1-Lucas(m)*x+(-1)^m*x^2+x*O(x^n))),n)} /* Paul D. Hanna, Jan 28 2012 */

Formula

G.f.: Sum_{n>=1} Fibonacci(n)*x^(n+1) / (1 - Lucas(n)*x + (-1)^n*x^2), where Lucas(n) = A000204(n). - Paul D. Hanna, Jan 28 2012

a(n) ~ c * phi^(n^2/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 1.14267253730874516106624178718900147373346430046702447860265114357421... - Vaclav Kotesovec, Jan 08 2021