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%I #22 Sep 08 2022 08:46:01
%S 1,5,21,70,214,614,1703,4619,12363,32812,86636,228012,598893,1571089,
%T 4118305,10790194,28262594,74014290,193807315,507451415,1328617751,
%U 3478516440,9107117016,23843134680,62422772569,163425968669,427856404653
%N a(n) = Sum_{j=1..n} Sum_{i=1..n} F(i,j), where F is the Fibonacci fusion array of A202453.
%C Partial sums of A188516.
%H G. C. Greubel, <a href="/A202462/b202462.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-4,10,-2,-3,1).
%F G.f.: x*(1+2*x^2-x^3)/((1+x)*(1-3*x+x^2)*(1-x-x^2)*(1-x)^2). - _R. J. Mathar_, Dec 20 2011
%F a(n) = Fibonacci(n+2)*Fibonacci(n+3) - 2*Fibonacci(n+4) + n + 4. - _G. C. Greubel_, Jul 23 2019
%t (* First program *)
%t n = 28;
%t Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
%t Table[Fibonacci[k], {k, 1, n}]];
%t P = Transpose[Q]; F = P.Q;
%t a[m_] := Sum[F[[i]][[j]], {i, 1, m}, {j, 1, m}]
%t Table[a[m], {m, 1, n}] (* A202462 *)
%t Table[a[m] - a[m - 1], {m, 1, n}] (* A188516 *)
%t (* Additional programs *)
%t LinearRecurrence[{5,-6,-4,10,-2,-3,1},{1,5,21,70,214,614,1703},30] (* _Harvey P. Dale_, Jul 23 2015 *)
%t With[{F=Fibonacci}, Table[F[n+2]*F[n+3] -2*F[n+4] +n+4, {n,30}]] (* _G. C. Greubel_, Jul 23 2019 *)
%o (PARI) vector(30, n, f=fibonacci; f(n+2)*f(n+3) -2*f(n+4) +n+4) \\ _G. C. Greubel_, Jul 23 2019
%o (Magma) F:=Fibonacci; [F(n+2)*F(n+3) -2*F(n+4) +n+4: n in [1..30]]; // _G. C. Greubel_, Jul 23 2019
%o (Sage) f=fibonacci; [f(n+2)*f(n+3)-2*f(n+4) +n+4 for n in (1..30)] # _G. C. Greubel_, Jul 23 2019
%o (GAP) F:=Fibonacci;; List([1..30], n-> F(n+2)*F(n+3) -2*F(n+4) +n+4); # _G. C. Greubel_, Jul 23 2019
%Y Cf. A000045, A188616, A202451.
%K nonn
%O 1,2
%A _Clark Kimberling_, Dec 19 2011
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