A053535 - OEIS

%I #21 Sep 08 2022 08:45:00

%S 1,6,63,540,4941,44226,398763,3586680,32286681,290560446,2615103063,

%T 23535750420,211822285221,1906398972666,17157595536963,

%U 154418345483760,1389765152400561,12507886242464886,112570976569604463

%N Expansion of 1/((1+3*x)*(1-9*x)).

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H G. C. Greubel, <a href="/A053535/b053535.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,27).

%F a(n) = (3^n/4)*(3^(n+1) + (-1)^n).

%F a(n) = 6*a(n-1) + 27*a(n-2), with a(0)=1, a(1)=6.

%F E.g.f.: (3*exp(9*x) + exp(-3*x))/4. - _G. C. Greubel_, May 16 2019

%t LinearRecurrence[{6,27}, {1,6}, 20] (* _G. C. Greubel_, May 16 2019 *)

%o (PARI) my(x='x+O('x^20)); Vec(1/((1+3*x)*(1-9*x))) \\ _G. C. Greubel_, May 16 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 1/((1+3*x)*(1-9*x)) )); // _G. C. Greubel_, May 16 2019

%o (Sage) (1/((1+3*x)*(1-9*x))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 16 2019

%o (GAP) a:=[1,6];; for n in [3..30] do a[n]:=6*a[n-1]+27*a[n-2]; od; a; # _G. C. Greubel_, May 16 2019

%Y Cf. A015518.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Jan 15 2000