A110111 Sequence associated to the recurrence b(n) = b(n-1) + 3*b(n-2).

0, 1, 7, 133, 1330, 18430, 210490, 2673223, 31940881, 394918819, 4788779380, 58709030380, 715296121540, 8745656280829, 106717441265323, 1303667366328817, 15915556720909510, 194371775990116810

Offset: 0

Author

Paul Barry, Jul 12 2005

Keywords

• easy
• nonn

Comments

In general, let T(n,k) be the solution to T(n,k) = T(n-1,k) + k*T(n-2,k) for n >= 2 with T(0,k) = 0 and T(1,k) = 1 for all k. Then, for fixed k, S(n,k) = T(n,k) * T(n+1,k) * T(n+2,k)/(k+1) has g.f. x/((1 + k*x - k^3*x^2) * (1 - (3*k + 1)*x - k^3*x^2)) (cf. A110112). For the current sequence, a(n) = S(n,k=3) = T(n,3) * T(n+1,3) * T(n+2,3)/4. [Edited by Petros Hadjicostas, Dec 26 2019]

Links

• Index entries for linear recurrences with constant coefficients, signature (7,84,-189,-729).

Crossrefs

Cf. A006130, A110112.

Programs

Mathematica

CoefficientList[Series[x/((1+3x-27x^2)(1-10x-27x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{7,84,-189,-729},{0,1,7,133},30] (* Harvey P. Dale, Jun 29 2022 *)

Formula

G.f.: x/((1 + 3*x - 27*x^2) * (1 - 10*x - 27*x^2)).

a(n) = 7*a(n-1) + 84*a(n-2) - 189*a(n-3) - 729*a(n-4) for n >= 4.

a(n) = b(n) * b(n+1) * b(n+2)/4, where b(n) = (((1 + sqrt(13))/2 )^n - ((1 - sqrt(13))/2)^n)/sqrt(13). [Corrected by Petros Hadjicostas, Dec 26 2019]

a(n) = A006130(n-1) * A006130(n) * A006130(n+1)/4 for n >= 1.