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

1,3

Links

Table of n, a(n) for n=1..22.

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

Mathematica

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

Prog

(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 */

Crossrefs

Cf. Antidiagonal sums of array A102310.

Cf. A000204, A001622.

Sequence in context: A150323 A150324 A150325 * A150326 A150327 A150328

Adjacent sequences: A102308 A102309 A102310 * A102312 A102313 A102314

Keyword

nonn

Author

Ralf Stephan, Jan 06 2005

Status

approved