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Expansion of 1/((1-8*x)*(1-9*x)*(1-12*x)).
1

%I #26 Sep 08 2022 08:44:45

%S 1,29,565,9245,137221,1915949,25677205,334395965,4265954341,

%T 53604494669,666045061045,8206250792285,100467119521861,

%U 1224084180206189,14859716922480085,179888148281902205

%N Expansion of 1/((1-8*x)*(1-9*x)*(1-12*x)).

%H Vincenzo Librandi, <a href="/A020978/b020978.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (29,-276,864).

%F a(n) = (12^(n+2) - 4*9^(n+2) + 3*8^(n+2))/12. - _Yahia Kahloune_, Jun 30 2013

%F a(0)=1, a(1)=29, a(2)=565; for n>2, a(n) = 29*a(n-1) -276*a(n-2) +864*a(n-3). - _Vincenzo Librandi_, Jul 05 2013

%F a(n) = 21*a(n-1) -108*a(n-2) +8^n. - _Vincenzo Librandi_, Jul 05 2013

%t CoefficientList[Series[1 / ((1 - 8 x) (1 - 9 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 05 2013 *)

%t LinearRecurrence[{29, -276, 864}, {1, 29, 565}, 16] (* _Robert G. Wilson v_, Jul 05 2013 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-8*x)*(1-9*x)*(1-12*x)))); /* or */ I:=[1, 29, 565]; [n le 3 select I[n] else 29*Self(n-1)-276*Self(n-2)+864*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 05 2013

%o (PARI) x='x+O('x^30); Vec(1/((1-8*x)*(1-9*x)*(1-12*x))) \\ _G. C. Greubel_, Feb 09 2018

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_