A121963 Expansion of x^2*(1 + 2*x + 7*x^2 - 3*x^3 + x^4)/(1 - 26*x^3 - x^6).

0, 1, 2, 7, 23, 53, 182, 599, 1380, 4739, 15597, 35933, 123396, 406121, 935638, 3213035, 10574743, 24362521, 83662306, 275349439, 634361184, 2178432991, 7169660157, 16517753305, 56722920072, 186686513521, 430095947114

Offset: 1

Author

Roger L. Bagula, Sep 02 2006

Keywords

• nonn
• easy

Comments

a(n) is a component of the n-th partial product of 2 X 2 matrices with rows (0,1), (1, 1 + A130196(j)), j>=1.

The linear recurrence shows that these are three interleaved sequences (0,7,182,...), (1,23,599,...) and (2,53,1380,...) obeying simple recurrences of the form b(n) = 26*b(n-1) + b(n-2).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,26,0,0,1).

Programs

GAP

a:=[1,2,9];; for n in [7..30] do a[n]:=26*a[n-3]+a[n-6]; od; a; # G. C. Greubel, Oct 05 2019

Magma

I:=[0,1,2,7,23,53]; [n le 6 select I[n] else 26*Self(n-3) +Self(n-6): n in [1..30]]; // G. C. Greubel, Oct 05 2019

Maple

seq(coeff(series(x^2*(1+2*x+7*x^2-3*x^3+x^4)/(1-26*x^3-x^6), x, n+1), x, n), n = 1..30); # G. C. Greubel, Oct 05 2019

Mathematica

M[n_] := {{0,1}, {1, 1+Mod[n^2-n-1, 3]} }; v[1] = {0,1}; v[n_] := v[n] = M[n].v[n-1]; Table[v[n][[1]], {n,30}]
Rest@CoefficientList[Series[x^2*(1+2*x+7*x^2-3*x^3+x^4)/(1-26*x^3-x^6), {x,0,30}], x] (* G. C. Greubel, Oct 05 2019 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x^2*(1+2*x+7*x^2-3*x^3 +x^4)/( 1-26*x^3-x^6))) \\ G. C. Greubel, Oct 05 2019

Sage

def A121963_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1+2*x+7*x^2-3*x^3+x^4)/(1-26*x^3-x^6) ).list()
a=A121963_list(30); a[1:] # G. C. Greubel, Oct 05 2019