A135343 a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).

1, 3, 12, 51, 205, 820, 3277, 13107, 52428, 209715, 838861, 3355444, 13421773, 53687091, 214748364, 858993459, 3435973837, 13743895348, 54975581389, 219902325555, 879609302220, 3518437208883, 14073748835533, 56294995342132, 225179981368525, 900719925474099

Offset: 0

Author

Paul Curtz, Dec 05 2007

Keywords

• nonn
• easy

Comments

See A129339 comments. Third sequence after b(n) = 3*b(n-1) - 3*b(n-2) + 2*b(n-3) and c(n) = 3*c(n-1) - c(n-3) + 3*c(n-4). The first is for every sequence identical to its third differences. What characterizes the two others?

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3, 4, -1, 3, 4).

Crossrefs

Cf. A129339, A135345.

Programs

Mathematica

LinearRecurrence[{3,4,-1,3,4},{1,3,12,51,205},30] (* Harvey P. Dale, Jun 03 2013 *)

PARI

Vec((1-x+4*x^3)/((1+x)*(1-4*x)*(1-x+x^2)) + O(x^30)) \\ Colin Barker, Oct 11 2016

Formula

a(n+1) - 4*a(n) = hexaperiodic -1, 0, 3, 1, 0, -3.

a(n) = (1/15)*( 3*4^(n+1) - 2*(-1)^n + 5*cos(Pi*n/3) - 5*sqrt(3)*cos(Pi*n/3) ). - Richard Choulet, Jan 04 2008

G.f.: (1-x+4*x^3) / ((1+x)*(1-4*x)*(1-x+x^2)). - Colin Barker, Oct 11 2016

Extensions

More terms from Harvey P. Dale, Jun 03 2013

Removed incorrect formula, Joerg Arndt, Oct 11 2016