%I #13 Oct 10 2025 05:36:45

%S 1,0,1,2,3,15,32,126,351,1094,3321,9801,29768,88452,266085,797162,

%T 2390391,7175547,21516800,64573362,193700403,581130734,1743421725,

%U 5230147077,15690706952,47071500840,141215033961,423644304722,1270932117003,3812799539655

%N Cogrowth sequence of the 18-element group S3 X C3 = <S,T,U | S^3, T^2, U^3, (ST)^2, [S,U], [T,U]>.

%C Also called: D3 X C3. Gap identifier 18, 3.

%H Vincenzo Librandi, <a href="/A378109/b378109.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-8,-12,24,9,9,-27).

%F G.f.: (3*x^8-3*x^6-16*x^5+6*x^4+7*x^3-2*x^2-3*x+1) / ((x-1) * (3*x-1) * (x^2+x+1) * (3*x^2+3*x+1) * (3*x^2-3*x+1)).

%t CoefficientList[Series[(3*x^8-3*x^6-16*x^5+6*x^4+7*x^3-2*x^2-3*x+1)/((x-1)*(3*x-1)*(x^2+x+1)*(3*x^2+3*x+1)*(3*x^2-3*x+1)),{x,0,30}],x] (* _Vincenzo Librandi_, Oct 09 2025 *)

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3*x^8-3*x^6-16*x^5+6*x^4+7*x^3-2*x^2-3*x+1) / ((x-1) * (3*x-1) * (x^2+x+1) * (3*x^2+3*x+1) * (3*x^2-3*x+1)))); // _Vincenzo Librandi_, Oct 09 2025

%Y Cf. A095364 (D9), A378031 (C6 X C3), A378110 (S3:C3), A001045 (S3).

%K nonn,easy

%O 0,4

%A _Sean A. Irvine_, Nov 16 2024