A227047 Expansion of x^2*(1+x^2) / ( (x^2-x+1)*(-x^2-x+1)*(1+x+x^2) ).

0, 0, 1, 1, 2, 3, 4, 7, 12, 19, 31, 50, 80, 130, 211, 341, 552, 893, 1444, 2337, 3782, 6119, 9901, 16020, 25920, 41940, 67861, 109801, 177662, 287463, 465124, 752587, 1217712, 1970299, 3188011, 5158310, 8346320, 13504630, 21850951, 35355581, 57206532, 92562113, 149768644

Offset: 0

Author

Paul Curtz, Jun 29 2013

Keywords

• nonn
• less
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,1).

Programs

Mathematica

CoefficientList[Series[x^2(1+x^2)/((x^2-x+1)(-x^2-x+1)(1+x+x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,0,1,0,1,1},{0,0,1,1,2,3},50] (* Harvey P. Dale, Oct 16 2018 *)

Formula

a(0)=a(1)=0. a(n+2)=a(n+1)+a(n) + A134667(n+1).

a(2n+1) = A182895(n). a(2n+2) = A182895(n+1)-A182895(n).

a(n+1)/a(n) tends to A001622 (the golden ratio) as n->infinity.

a(n) = A079962(n-2) + A079962(n-4). - R. J. Mathar, Jun 30 2013

a(n+6) - a(n-6) = 10*A000045(n).

a(n+3) - a(n-3) = A000032(n).

a(n) = a(n-1) +a(n-3) +a(n-5) +a(n-6). - Joerg Arndt, Jun 30 2013