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

1,2

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).

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

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A213566, A213500.

Sequence in context: A244245 A288116 A165394 * A271908 A257448 A288415

Adjacent sequences: A213567 A213568 A213569 * A213571 A213572 A213573

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved