A213567 Principal diagonal of the convolution array A213566.

1, 13, 59, 183, 476, 1108, 2409, 4993, 10007, 19559, 37504, 70832, 132145, 244029, 446763, 811847, 1465676, 2630836, 4697945, 8350305, 14779671, 26058903, 45784224, 80179968, 139995361, 243755533, 423324539, 733409943

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 (6,-12,5,12,-12,-3,6,0,-1).

Crossrefs

Cf. A213566, A213500.

Programs

GAP

F:=Fibonacci;; List([1..30], n-> (2*n+3)*F(n+3)+(n^2+2)*F(n+2) -4*(n^2+2*n+2)); # G. C. Greubel, Jul 26 2019

Magma

F:= Fibonacci; [(2*n+3)*F(n+3)+(n^2+2)*F(n+2) -4*(n^2+2*n+2): n in [1..30]]; // G. C. Greubel, Jul 26 2019

Mathematica

(* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= n^2;
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}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213566 *)
d = Table[t[n, n], {n, 1, 40}] (* A213567 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213570 *)
(* Second program *)
Table[(2*n+3)*Fibonacci[n+3] +(n^2+2)*Fibonacci[n+2] -4*(n^2+2*n+2), {n, 30}] (* G. C. Greubel, Jul 26 2019 *)

PARI

vector(30, n, f=fibonacci; (2*n+3)*f(n+3)+(n^2+2)*f(n+2) -4*(n^2+ 2*n+2)) \\ G. C. Greubel, Jul 26 2019

Sage

f=fibonacci; [(2*n+3)*f(n+3)+(n^2+2)*f(n+2) -4*(n^2+ 2*n+2) for n in (1..30)] # G. C. Greubel, Jul 26 2019

Formula

a(n) = 6*a(n-1) - 12*a(n-2) - 5*a(n-3) + 12*a(n-4) - 12*a(n-5) - 3*a(n-6) + 6*a(n-7) - a(n-9).

G.f.: f(x)/g(x), where f(x) = x*(1 + 7*x - 7*x^2 - 20*x^3 + 9*x^4 + 9*x^5 + 9*x^6) and g(x) = (1 - 2*x + x^3)^3.

a(n) = (2*n + 3)*Fibonacci(n+3) + (n^2 + 2)*Fibonacci(n+2) - 4*(n^2 + 2*n + 2). - G. C. Greubel, Jul 26 2019