A167896 - OEIS

%I #14 Dec 07 2024 01:59:35

%S 1,5,20,80,320,1280,5120,20480,81920,327680,1310720,5242880,20971520,

%T 83886080,335544320,1342177280,5368709110,21474836400,85899345450,

%U 343597381200,1374389522400,5497558080000,21990232281600,87960928972800

%N Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A003947, 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="/A167896/b167896.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,-6).

%F G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*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)/ ( 6*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

%F From _G. C. Greubel_, Dec 06 2024: (Start)

%F a(n) = 3*Sum_{j=1..15} a(n-j) - 6*a(n-16).

%F G.f.: (1+x)*(1-x^16)/(1 - 4*x + 9*x^16 - 6*x^17). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-4*t+9*t^16-6*t^17), {t,0,50}], t] (* _G. C. Greubel_, Jul 01 2016; Dec 06 2024 *)

%t coxG[{16,6,-3,40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 06 2024 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-4*x+9*x^16-6*x^17) )); // _G. C. Greubel_, Dec 06 2024

%o (SageMath)

%o def A167896_list(prec):

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

%o     return P( (1+x)*(1-x^16)/(1-4*x+9*x^16-6*x^17) ).list()

%o print(A167896_list(40)) # _G. C. Greubel_, Dec 06 2024

%Y Cf. A003947, A154639, A169452.

%K nonn

%O 0,2

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