%I #20 Sep 08 2022 08:45:27

%S 0,1,2,7,23,53,182,599,1380,4739,15597,35933,123396,406121,935638,

%T 3213035,10574743,24362521,83662306,275349439,634361184,2178432991,

%U 7169660157,16517753305,56722920072,186686513521,430095947114

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

%C 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.

%C 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).

%H G. C. Greubel, <a href="/A121963/b121963.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,26,0,0,1).

%p 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

%t 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}]

%t 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 *)

%o (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

%o (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

%o (Sage)

%o def A121963_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( x^2*(1+2*x+7*x^2-3*x^3+x^4)/(1-26*x^3-x^6) ).list()

%o a=A121963_list(30); a[1:] # _G. C. Greubel_, Oct 05 2019

%o (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

%K nonn,easy

%O 1,3

%A _Roger L. Bagula_, Sep 02 2006