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%I #30 Nov 19 2017 14:31:35
%S 0,0,0,1,1,6,12,42,107,323,888,2568,7224,20629,58429,166230,471780,
%T 1340730,3807431,10816631,30722736,87272592,247895472,704164537,
%U 2000191753,5681637318,16138865148,45843078954,130218850259
%N Expansion of x^3 / (1 - x - 5*x^2 - x^3 + x^4).
%H Li Zhou et al., <a href="https://www.jstor.org/stable/27642267">Tiling 4-Rowed Rectangles with Dominoes: Problem 11187</a>, American Mathematical Monthly, vol. 114, no. 6, 2007 (pp. 554-556).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,1,-1).
%F a(n) = a(n - 1) + 5*a(n - 2) + a(n - 3) - a(n - 4).
%F a(n) = ((-b + c - e - g + i)*(1 + s + k)^(n - 1) + ( b + d - f + h - j)*(1 - s + l)^(n - 1) + ( b - d + f - h + j)*(1 - s - l)^(n - 1) + (-b - c + e + g - i)*(1 + s - k)^(n - 1))/(5800*4^(n - 1)), 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
%t 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]
%o (PARI) concat(vector(3), Vec(x^3/(1-x-5*x^2-x^3+x^4) + O(x^30))) \\ _Michel Marcus_, Nov 19 2017
%Y Cf. A005178.
%K nonn,easy
%O 0,6
%A _Roger L. Bagula_, Jun 07 2007