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Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
7

%I #14 Dec 04 2024 05:51:01

%S 1,11,110,1100,11000,110000,1100000,11000000,110000000,1100000000,

%T 11000000000,110000000000,1100000000000,11000000000000,

%U 110000000000000,1100000000000000,10999999999999945,109999999999998900

%N Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A003953, 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="/A167914/b167914.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 (9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,-45).

%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)/( 45*t^16 - 9*t^15 - 9*t^14 - 9*t^13 - 9*t^12 - 9*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

%F From _G. C. Greubel_, Dec 04 2024: (Start)

%F a(n) = 9*Sum_{j=1..15} a(n-j) - 45*a(n-16).

%F G.f.: (1+x)*(1-x^16)/(1 - 10*x + 54*x^16 - 45*x^17). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-10*t+54*t^16-45*t^17), {t,0,50}], t] (* _G. C. Greubel_, Jul 01 2016; Dec 04 2024 *)

%t coxG[{16,45,-9}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 04 2024 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 40);

%o Coefficients(R!( (1+x)*(1-x^16)/(1-10*x+54*x^16-45*x^17) )); // _G. C. Greubel_, Dec 04 2024

%o (SageMath)

%o def A167914_list(prec):

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

%o return P( (1+x)*(1-x^16)/(1-10*x+54*x^16-45*x^17) ).list()

%o A167914_list(40) # _G. C. Greubel_, Dec 04 2024

%Y Cf. A003953, A154638, A169452.

%K nonn

%O 0,2

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