A191272 Expansion of x*(4+5*x)/( (1-4*x)*(1 + x + x^2) ).

0, 4, 17, 63, 256, 1025, 4095, 16384, 65537, 262143, 1048576, 4194305, 16777215, 67108864, 268435457, 1073741823, 4294967296, 17179869185, 68719476735, 274877906944, 1099511627777, 4398046511103, 17592186044416, 70368744177665

Offset: 0

Author

Paul Curtz, May 29 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..175
• Index entries for linear recurrences with constant coefficients, signature (3,3,4).

Programs

Magma

m:=24; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(4+5*x)/((1-4*x)*(1+x+x^2))));  // Bruno Berselli, Jul 04 2011

Mathematica

LinearRecurrence[{3,3,4},{0,4,17},30] (* or *) CoefficientList[ Series[ x (4+5x)/((1-4x)(1+x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Jun 19 2011 *)

Maxima

makelist(coeff(taylor(x*(4+5*x)/((1-4*x)*(1+x+x^2)), x, 0, n), x, n), n, 0, 23);  /* Bruno Berselli, Jun 06 2011 */

PARI

a(n)=4^n-[1,0,-1][n%3+1] \\ Charles R Greathouse IV, Jun 06 2011

Formula

G.f.: x*(4+5*x)/(1 - 3*x - 3*x^2 - 4*x^3).

a(n) = 4^n-A057078(n) = 4^n - (n-th element of periodic length 3 repeat 1, 0, -1)

a(n) = A024495(2*n) + A024495(1+2*n).

a(n+1) = 4*a(n) + (n-th element of periodic length 3 repeat 4, 1, -5).

a(n) = A052539(n) - A080425(n+1). - Bruno Berselli, Jun 06 2011

a(0)=0, a(1)=4, a(2)=17, a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3). - Harvey P. Dale, Jun 19 2011

a(n) = 4*a(n-1) + a(n-3) - 4*(n-4) (n>3). - Bruno Berselli, Jul 04 2011