A129113 Expansion of x^3 / (1 - x - 5*x^2 - x^3 + x^4).

0, 0, 0, 1, 1, 6, 12, 42, 107, 323, 888, 2568, 7224, 20629, 58429, 166230, 471780, 1340730, 3807431, 10816631, 30722736, 87272592, 247895472, 704164537, 2000191753, 5681637318, 16138865148, 45843078954, 130218850259

Offset: 0

Author

Roger L. Bagula, Jun 07 2007

Keywords

• nonn
• easy

Links

• Li Zhou et al., Tiling 4-Rowed Rectangles with Dominoes: Problem 11187, American Mathematical Monthly, vol. 114, no. 6, 2007 (pp. 554-556).
• Index entries for linear recurrences with constant coefficients, signature (1,5,1,-1).

Crossrefs

Cf. A005178.

Programs

Mathematica

f[1] = f[2] = f[3] = 0; f[4] = 1; f[n_] := f[n] = f[n - 1] + 5f[n - 2] + f[n - 3] - f[n - 4]; Array[f, 29] (* or *) LinearRecurrence[{1, 5, 1, -1}, {0, 0, 0, 1}, 29] (* or *) gf = x^3/(1 - x - 5 x^2 - x^3 + x^4); CoefficientList[ Series[gf, {x, 0, 28}], x]

PARI

concat(vector(3), Vec(x^3/(1-x-5*x^2-x^3+x^4) + O(x^30))) \\ Michel Marcus, Nov 19 2017

Formula

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

a(n) = ((-b + c - e - g + i)*(1 + s + k)^n + (b + d - f + h - j)*(1 - s + l)^n + (b - d + f - h + j)*(1 - s - l)^n + (-b - c + e + g - i)*(1 + s - k)^n)/(5800*4^n), with b = 100*s, c = 1015*k, d = 145*sqrt(10*(7 + s)), e = 245*sqrt(58*(7 + s)), f = 75*sqrt(290*(7 + s)), g = 1914*sqrt(119 + 22*s), h = 98*sqrt(145*(119 + 22*s)), i = 382*sqrt(3451 + 638*s), j = 406*sqrt(595 + 110*s), k = sqrt(2*(7 + s)), l = sqrt(2*(7 - s)), s = sqrt(29). - Tim Monahan, Sep 09 2011; modified by Robert G. Wilson v, Sep 26 2011