A213557 Antidiagonal sums of the convolution array A213590.

1, 6, 23, 70, 184, 438, 971, 2042, 4125, 8076, 15424, 28876, 53189, 96670, 173747, 309362, 546456, 958690, 1672015, 2901170, 5011321, 8621976, 14781888, 25263000, 43053769, 73186038, 124119311, 210055582, 354806200, 598245006

Offset

1,2

Links

Clark Kimberling, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (5,-8,2,6,-4,-1,1).

Formula

a(n) = 5*a(n-1) - 8*a(n-2) + 2*a(n-3) + 6*a(n-4) - 4*a(n-5) - a(n-6) + a(n-7).

G.f.: f(x)/g(x), where f(x) = x*(1 + x + x^2 + x^3) and g(x) = (1 - x)^3 (1 - x - x^2)^2.

a(n) = n*Fibonacci(n+7) - 2*Fibonacci(n+9) + 2*n^2 + 20*n + 68. - G. C. Greubel, Jul 06 2019

Mathematica

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[n];
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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
With[{F = Fibonacci}, Table[n*F[n+7] -2*F[n+9] +2*(n^2+10*n+34), {n, 40}]] (* G. C. Greubel, Jul 06 2019 *)

Prog

(PARI) vector(40, n, f=fibonacci; n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34)) \\ G. C. Greubel, Jul 06 2019
(Magma) F:=Fibonacci; [n*F(n+7) -2*F(n+9) +2*(n^2+10*n+34): n in [1..40]]; // G. C. Greubel, Jul 06 2019
(Sage) f=fibonacci; [n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34) for n in (1..40)] # G. C. Greubel, Jul 06 2019
(GAP) F:=Fibonacci;; List([1..40], n-> n*F(n+7) -2*F(n+9) +2*(n^2+10*n+ 34)) # G. C. Greubel, Jul 06 2019

Crossrefs

Cf. A213590, A213500.

Sequence in context: A281424 A005745 A332081 * A273386 A045618 A038737

Adjacent sequences: A213554 A213555 A213556 * A213558 A213559 A213560

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved