A121991 a(n) = 3*a(n-1) - a(n-2) - a(n-3) + 12.

0, 1, 13, 50, 148, 393, 993, 2450, 5976, 14497, 35077, 84770, 204748, 494409, 1193721, 2882018, 6957936, 16798081, 40554301, 97906898, 236368324, 570643785, 1377656145, 3325956338, 8029569096, 19385094817

Offset: 0

Author

Roger L. Bagula, Sep 10 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4, -4, 0, 1).

Crossrefs

Cf. A000129, A003215, A005891.

Programs

Mathematica

RecurrenceTable[{a[n] == 3*a[n - 1] - a[n - 2] - a[n - 3] + 12, a[0] == 0, a[1] == 1, a[2] == 13}, a, {n,0,50}] (* or *) LinearRecurrence[{4,-4,0,1}, {0,1,13,50}, 50] (* G. C. Greubel, Sep 14 2017 *)

PARI

x='x+O('x^50); concat([0], Vec(-x(1+9x+2x^2)/((1-x)^2*(x^2+2x-1))) ) \\ G. C. Greubel, Sep 14 2017

Formula

a(n) = ((11 - 7*sqrt(2))*(1 - sqrt(2))^n + (1 + sqrt(2))^n*(11 + 7*sqrt(2)) - 24*n - 22)/4.

O.g.f.: -x(1+9x+2x^2)/((1-x)^2*(x^2+2x-1)). - R. J. Mathar, Aug 22 2008

a(n) = -6(n+1)+(1+11*A000129(n+1)+3*A000129(n))/2. - R. J. Mathar, Aug 22 2008

E.g.f.: (1/2)*(11*cosh(sqrt(2)*x) + 7*sqrt(2)*sinh(sqrt(2)*x) - (12*x + 11))*exp(x). - G. C. Greubel, Sep 14 2017

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4). - Wesley Ivan Hurt, May 04 2024

Extensions

Edited by N. J. A. Sloane, Aug 24 2008, Dec 30 2008