A213570 Antidiagonal sums of the convolution array A213566.

1, 9, 37, 110, 272, 598, 1213, 2323, 4265, 7588, 13184, 22500, 37881, 63125, 104381, 171602, 280896, 458330, 746085, 1212415, 1967761, 3190824, 5170752, 8375400, 13561777, 21954753, 35536213, 57512918, 93073520, 150613438

Offset: 1

Author

Clark Kimberling, Jun 19 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 (5,-9,6,1,-3,1).

Crossrefs

Cf. A213566, A213500.

Programs

GAP

List([1..35], n-> Fibonacci(n+9)+Lucas(1,-1,n+8)[2] -(n^3+9*n^2 +39*n+81)); # G. C. Greubel, Jul 26 2019

Magma

[Fibonacci(n+9) +Lucas(n+8) -(n^3+9*n^2+39*n+81): n in [1..35]]; // 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[Fibonacci[n+9] + LucasL[n+8] -(n^3+9*n^2+39*n+81), {n,35}] (* G. C. Greubel, Jul 26 2019 *)

PARI

vector(35,n, f=fibonacci; 2*f(n+9)+f(n+7) -(n^3+9*n^2+39*n+81)) \\ G. C. Greubel, Jul 26 2019

Sage

[fibonacci(n+9) +lucas_number2(n+8,1,-1) -(n^3+9*n^2+39*n+81) for n in (1..35)] # G. C. Greubel, Jul 26 2019

Formula

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

G.f.: f(x)/g(x), where f(x) = x*(1 + 4*x + x^2) and g(x) = (1 - x - x^2)*(1 - x)^4.

a(n) = Fibonacci(n+9) + Lucas(n+8) - n*(n^2 + 9*n + 39) - 81. - Ehren Metcalfe, Jul 10 2019

a(n) = Sum_{k=1..n} k^3 * Fibonacci(n+1-k). - Greg Dresden, Feb 27 2022