A017918 - OEIS

%I #20 Sep 08 2022 08:44:43

%S 1,20,289,3740,46321,563300,6797569,81762860,982121041,11790305780,

%T 141507994849,1698217742780,20379222467761,244553718979460,

%U 2934659879368129,35215994824839500,422592319303230481

%N Expansion of 1/((1-3x)(1-5x)(1-12x)).

%H Vincenzo Librandi, <a href="/A017918/b017918.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (20,-111,180).

%F a(0)=1, a(1)=20, a(2)=289; for n>2, a(n) = 20*a(n-1) -111*a(n-2) +180*a(n-3). - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = 17*a(n-1) -60*a(n-2) +3^n. - _Vincenzo Librandi_, Jul 02 2013

%F a(n) = (2*12^(n+2) - 9*5^(n+2) + 7*3^(n+2))/126. [_Yahia Kahloune_, Jul 06 2013]

%p a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [20, -111, 180][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 02 2013

%t CoefficientList[Series[1 / ((1 - 3 x) (1 - 5 x) (1 - 12 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 02 2013 *)

%t LinearRecurrence[{20,-111,180},{1,20,289},20] (* _Harvey P. Dale_, Aug 22 2019 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-5*x)*(1-12*x)))); /* or */ I:=[1, 20, 289]; [n le 3 select I[n] else 20*Self(n-1)-111*Self(n-2)+180*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 02 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_.