A108014 Expansion of (x^2-2*x)/(x^4-x^2+2*x-1).

0, 2, 3, 4, 5, 8, 14, 24, 39, 62, 99, 160, 260, 422, 683, 1104, 1785, 2888, 4674, 7564, 12239, 19802, 32039, 51840, 83880, 135722, 219603, 355324, 574925, 930248, 1505174, 2435424, 3940599, 6376022, 10316619, 16692640, 27009260, 43701902

Offset: 0

Author

Roger L. Bagula, May 30 2005

Keywords

• nonn
• easy

Links

• Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Programs

Mathematica

M = {{0, 0, 0, 1}, {1, 1, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 1}} v[1] = {0, 1, 1, 2}; v[n_] := v[n] = M.v[n - 1]; digits = 50; a = Table[v[n][[1]], {n, 1, digits}]
CoefficientList[Series[(x^2-2x)/(x^4-x^2+2x-1),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,0,1},{0,2,3,4},41] (* Harvey P. Dale, Aug 19 2011 *)

PARI

Vec((x^2-2*x)/(x^4-x^2+2*x-1)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

M = {{0, 0, 0, 1}, {1, 1, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 1}} v[n]=v[n-1].M a(n) = v[n][[1]].

a(n) = (1/2)*[Lucas(n+1) + 2*cos((n+4)*Pi/3)].

a(n) = (1/2)*[A000032(n+1) + A057079(n+5)].

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