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

1,2

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).

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

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A213566, A213500.

Sequence in context: A171749 A141917 A163833 * A124864 A126400 A297962

Adjacent sequences: A213564 A213565 A213566 * A213568 A213569 A213570

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved