A166413 - OEIS

%I #22 Jun 08 2026 10:22:10

%S 1,19,342,6156,110808,1994544,35901792,646232256,11632180608,

%T 209379250944,3768826516992,67838877305685,1221099791499252,

%U 21979796246931303,395636332443769260,7121453983969951188

%N Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%C The initial terms coincide with those of A170738, 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="/A166413/b166413.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 (17,17,17,17,17,17,17,17,17,17,-153).

%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)/(153*t^11 - 17*t^10 - 17*t^9 - 17*t^8 - 17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 - 17*t + 1).

%F From _G. C. Greubel_, Jul 23 2024: (Start)

%F a(n) = 17*Sum_{j=1..10} a(n-j) - 153*a(n-11).

%F G.f.: (1+x)*(1-x^11)/(1 - 18*x + 170*x^11 - 153*x^12). (End)

%t With[{p=153, q=17}, 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 12 2016; Jul 23 2024 *)

%t coxG[{11,153,-17}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Nov 26 2022 *)

%o (Magma)

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

%o Coefficients(R!( (1+x)*(1-x^11)/(1-18*x+170*x^11-153*x^12) )); // _G. C. Greubel_, Jul 23 2024

%o (SageMath)

%o def A166413_list(prec):

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

%o     return P( (1+x)*(1-x^11)/(1-18*x+170*x^11-153*x^12) ).list()

%o A166413_list(30) # _G. C. Greubel_, Jul 23 2024

%o (PARI) Vec((1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)*(1+x)/(1-17*x-17*x^2-17*x^3-17*x^4-17*x^5-17*x^6-17*x^7-17*x^8-17*x^9-17*x^10+153*x^11)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 08 2026

%Y Cf. A154638, A169452, A170738.

%K nonn,easy

%O 0,2

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