%I #14 Mar 18 2023 23:04:30

%S 0,1,2,7,22,86,414,2521,19494,191695,2397716,38148444,772057396,

%T 19875413009,650843469738,27110077916903,1436411242814058,

%U 96810095832996034,8299583912379548210,905077596297808256825,125547805293905152853710,22152679283963321048140511

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

%F 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

%F 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

%t Table[Sum[Fibonacci[k(n-k)],{k,n}],{n,30}] (* _Harvey P. Dale_, Jul 03 2019 *)

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

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {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 */

%Y Cf. Antidiagonal sums of array A102310.

%Y Cf. A000204, A001622.

%K nonn

%O 1,3

%A _Ralf Stephan_, Jan 06 2005