A213588 Principal diagonal of the convolution array A213587.

1, 7, 27, 96, 315, 994, 3043, 9123, 26909, 78370, 225911, 645732, 1832677, 5170111, 14509695, 40537284, 112805043, 312808198, 864707719, 2383649115, 6554153921, 17980221382, 49222822127, 134495771976, 366850762825

Offset: 1

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,5,-1).

Crossrefs

Cf. A213587, A213500.

Programs

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

Magma

[(n*Lucas(2*n+2) - Fibonacci(n)*Lucas(n-1))/5: n in [1..30]]; // G. C. Greubel, Jul 08 2019

Mathematica

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+1];
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[(n*LucasL[2n+2] -Fibonacci[n]*LucasL[n-1])/5, {n, 30}] (* G. C. Greubel, Jul 08 2019 *)

PARI

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
vector(30, n, (n*lucas(2*n+2) - fibonacci(n)*lucas(n-1))/5) \\ G. C. Greubel, Jul 08 2019

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

Formula

a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 5*a(n-4) - a(n-5).

G.f.: x*(1 + 2*x - 3*x^2 + x^3)/((1 + x)*(1 - 3*x + x^2)^2).

a(n) = (n*Lucas(2*n+2) - Fibonacci(n)*Lucas(n-1))/5. - G. C. Greubel, Jul 08 2019