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a(n)=sum_{j=0..n} sum_{i=0..j} F(i)*L(j), where F(n)=A000045(n) and L(n)=A000032(n).
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%I #29 Oct 03 2020 15:17:43

%S 0,1,7,23,72,204,564,1521,4059,10747,28336,74504,195576,512865,

%T 1344063,3521007,9221688,24148468,63230860,165555665,433454835,

%U 1134839091,2971111392,7778574288,20364739632,53315851969,139583151799,365434146311,956720165544

%N a(n)=sum_{j=0..n} sum_{i=0..j} F(i)*L(j), where F(n)=A000045(n) and L(n)=A000032(n).

%H Harvey P. Dale, <a href="/A242496/b242496.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-6,4,2,-1).

%F a(n) = A001519(n+2) - A000032(n+2) + A059841(n).

%F a(n) = L(n)*F(n+3) - L(n+2) + (1-3*(-1)^n)/2. - _Colin Barker_, May 18 2014

%F G.f.: -x*(3*x^2-3*x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - _Colin Barker_, May 16 2014

%e For n=5, 0*(2+1+3+4+7+11) + 1*(1+3+4+7+11) + 1*(3+4+7+11) + 2*(4+7+11) + 3*(7+11) + 5*11 = 204 = F(2*5+3) - L(n+2) + 0 = 233-29 = 204.

%p A242496 := proc(n)

%p add(add(A000045(i)*A000032(j),i=0..j),j=0..n) ;

%p end proc: # _R. J. Mathar_, May 17 2014

%t LinearRecurrence[{4,-2,-6,4,2,-1},{0,1,7,23,72,204},30] (* _Harvey P. Dale_, Oct 03 2020 *)

%o (PARI)

%o F(n) = fibonacci(n)

%o L(n) = if(n==0, 2, F(2*n)/F(n))

%o vector(30, n, sum(i=0, n-1, sum(j=i, n-1, F(i)*L(j)))) \\ _Colin Barker_, May 16 2014

%Y Cf. A190173, A000045, A000032, A242300.

%K nonn,easy

%O 0,3

%A _J. M. Bergot_, May 16 2014

%E Two terms corrected, and more terms added by _Colin Barker_, May 16 2014

%E Formula corrected by _Colin Barker_, May 17 2014