%I #17 Dec 03 2025 17:11:31

%S 1,28,756,20412,551124,14880348,401769396,10847773692,292889889684,

%T 7908027021468,213516729579636,5764951698649794,155653695863534232,

%U 4202649788315149080,113471544284501595192,3063731695681342461048

%N Number of reduced words of length n in Coxeter group on 28 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%C The initial terms coincide with those of A170747, although the two sequences are eventually different.

%C Computed with Magma using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A166422/b166422.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (26,26,26,26,26,26,26,26,26,26,-351).

%F G.f.: (t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(351*t^11 - 26*t^10 - 26*t^9 - 26*t^8 - 26*t^7 - 26*t^6 - 26*t^5 - 26*t^4 - 26*t^3 - 26*t^2 - 26*t + 1).

%F From _G. C. Greubel_, Jul 25 2024: (Start)

%F a(n) = 26*Sum_{j=1..10} a(n-j) - 351*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 27*x + 377*x^11 - 351*x^12). (End)

%t With[{p=351, q=26}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t,0,40}], t]] (* _G. C. Greubel_, May 13 2016; Jul 25 2024 *)

%t coxG[{11,351,-26}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Dec 22 2019 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 30);

%o Coefficients(R!( (1+x)*(1-x^11)/(1-27*x+377*x^11-351*x^12) )); // _G. C. Greubel_, Jul 25 2024

%o (SageMath)

%o def A166422_list(prec):

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

%o return P( (1+x)*(1-x^11)/(1-27*x+377*x^11-351*x^12) ).list()

%o A166422_list(30) # _G. C. Greubel_, Jul 25 2024

%Y Cf. A154638, A169452, A170747.

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009