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A213588 Principal diagonal of the convolution array A213587.

%I A213588 #18 Jan 09 2025 14:51:38
%S A213588 1,7,27,96,315,994,3043,9123,26909,78370,225911,645732,1832677,
%T A213588 5170111,14509695,40537284,112805043,312808198,864707719,2383649115,
%U A213588 6554153921,17980221382,49222822127,134495771976,366850762825
%N A213588 Principal diagonal of the convolution array A213587.
%H A213588 Clark Kimberling, <a href="/A213588/b213588.txt">Table of n, a(n) for n = 1..1000</a>
%H A213588 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-5,5,-1).
%F A213588 a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 5*a(n-4) - a(n-5).
%F A213588 G.f.: x*(1 + 2*x - 3*x^2 + x^3)/((1 + x)*(1 - 3*x + x^2)^2).
%F A213588 a(n) = (n*Lucas(2*n+2) - Fibonacci(n)*Lucas(n-1))/5. - _G. C. Greubel_, Jul 08 2019
%t A213588 (* First program *)
%t A213588 b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+1];
%t A213588 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213588 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213588 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213587 *)
%t A213588 r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
%t A213588 Table[T[n, n], {n, 1, 40}] (* A213588 *)
%t A213588 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213588 Table[s[n], {n, 1, 50}] (* A213589 *)
%t A213588 (* Second program *)
%t A213588 Table[(n*LucasL[2n+2] -Fibonacci[n]*LucasL[n-1])/5, {n, 30}] (* _G. C. Greubel_, Jul 08 2019 *)
%o A213588 (PARI) lucas(n) = fibonacci(n+1) + fibonacci(n-1);
%o A213588 vector(30, n, (n*lucas(2*n+2) - fibonacci(n)*lucas(n-1))/5) \\ _G. C. Greubel_, Jul 08 2019
%o A213588 (Magma) [(n*Lucas(2*n+2) - Fibonacci(n)*Lucas(n-1))/5: n in [1..30]]; // _G. C. Greubel_, Jul 08 2019
%o A213588 (Sage) [(n*lucas_number2(2*n+2,1,-1) - fibonacci(n)*lucas_number2(n-1, 1, -1))/5 for n in (1..30)] # _G. C. Greubel_, Jul 08 2019
%o A213588 (GAP) List([1..30], n-> (n*Lucas(1,-1,2*n+2)[2] - Fibonacci(n)*Lucas(1,-1,n-1)[2])/5); # _G. C. Greubel_, Jul 08 2019
%Y A213588 Cf. A213587, A213500.
%K A213588 nonn,easy
%O A213588 1,2
%A A213588 _Clark Kimberling_, Jun 19 2012