A213578 Antidiagonal sums of the convolution array A213576.

1, 4, 13, 34, 80, 174, 359, 712, 1371, 2580, 4768, 8684, 15629, 27852, 49225, 86390, 150704, 261530, 451795, 777360, 1332791, 2277864, 3882048, 6599064, 11191705, 18940564, 31992709, 53943562, 90807056, 152631750, 256190783

Offset: 1

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1).

Crossrefs

Cf. A213576, A213500.

Programs

GAP

List([1..40], n-> n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5)); # G. C. Greubel, Jul 05 2019

Magma

[n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5): n in [1..40]]; // Vincenzo Librandi, Jul 05 2019

Mathematica

b[n_]:= n; 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}]] (* A213576 *)
r[n_] := Table[t[n, k], {k,40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n,1,40}] (* A213577 *)
s[n_] := Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* alternate program *)
LinearRecurrence[{4,-4,-2,4,0,-1},{1,4,13,34,80,174},40] (* Harvey P. Dale, Jul 04 2019 *)

PARI

vector(40, n, n*fibonacci(n+4)-2*(fibonacci(n+5)-n-5)) \\ G. C. Greubel, Jul 05 2019

Sage

[n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5) for n in (1..40)] # G. C. Greubel, Jul 05 2019

Formula

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

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

a(n) = n*F(n+4) - 2*(F(n+5) - n - 5), F = A000045. - Ehren Metcalfe, Jul 05 2019