A135450 a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).

0, 0, 0, 1, 4, 16, 63, 252, 1008, 4033, 16132, 64528, 258111, 1032444, 4129776, 16519105, 66076420, 264305680, 1057222719, 4228890876, 16915563504, 67662254017, 270649016068, 1082596064272, 4330384257087, 17321537028348

Offset: 0

Author

Paul Curtz, Dec 14 2007

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 (4,0,-1,4).

Crossrefs

Cf. A135343, A135345.

Programs

Mathematica

a = {0, 0, 0, 1, 4}; Do[AppendTo[a, 3*a[[ -1]] + 4*a[[ -2]] - a[[ -3]] + 3*a[[ -4]] + 4*a[[ -5]]], {25}]; a (* Stefan Steinerberger, Dec 31 2007 *)
LinearRecurrence[{3, 4, -1, 3, 4}, {0, 0, 0, 1, 4}, 25] (* G. C. Greubel, Oct 14 2016 *)
LinearRecurrence[{4,0,-1,4},{0,0,0,1},40] (* Harvey P. Dale, Jan 31 2021 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 4,-1,0,4]^n*[0;0;0;1])[1,1] \\ Charles R Greathouse IV, Oct 14 2016

Formula

a(n+1) - 4*a(n) = hexaperiodic 0, 0, 1, 0, 0, -1, A131531.

a(n) + a(n+3) = 1, 4, 16, 64 = 2^2n = A000302.

a(n) = (1/65)*4^n + (1/15)*(-1)^(n+1) + (2/39)*cos((Pi*n)/3) - (4*sqrt(3)/39) * sin((Pi*n)/3). Or, a(n) = (1/65)*(4^n + [ -1; -4; -16; 1; 4; 16]). - Richard Choulet, Dec 31 2007

O.g.f.: -x^3/[(4*x-1)*(1+x)*(x^2-x+1)]. - R. J. Mathar, Jan 07 2008

Extensions

More terms from Stefan Steinerberger, Dec 31 2007