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Number of reduced words of length n in Coxeter group on 46 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
1

%I #19 Sep 08 2022 08:45:46

%S 1,46,2070,93150,4190715,188535600,8482007160,381596054400,

%T 17167581467190,772350369021000,34747182860785560,1563237055602189000,

%U 70328294002955286540,3163991615757072698400,142344458748855549948960

%N Number of reduced words of length n in Coxeter group on 46 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%C The initial terms coincide with those of A170765, 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="/A163232/b163232.txt">Table of n, a(n) for n = 0..600</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (44, 44, 44, -990).

%F G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(990*t^4 - 44*t^3 - 44*t^2 - 44*t + 1).

%F a(n) = 44*a(n-1)+44*a(n-2)+44*a(n-3)-990*a(n-4). - _Wesley Ivan Hurt_, May 10 2021

%t CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(990*t^4-44*t^3-44*t^2 - 44*t+1), {t,0,20}], t] (* or *) Join[{1}, LinearRecurrence[ {44, 44, 44, -990}, {46,2070,93150,4190715}, 20]] (* _G. C. Greubel_, Dec 11 2016 *)

%t coxG[{4, 990, -44}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, May 01 2019 *)

%o (PARI) my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(990*t^4-44*t^3 - 44*t^2-44*t+1)) \\ _G. C. Greubel_, Dec 11 2016

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^4)/(1-45*x+1034*x^4-990*x^5) )); // _G. C. Greubel_, May 01 2019

%o (Sage) ((1+x)*(1-x^4)/(1-45*x+1034*x^4-990*x^5)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 01 2019

%o (GAP) a:=[46,2070,93150,4190715];; for n in [5..20] do a[n]:=44*(a[n-1] +a[n-2] +a[n-3]) -990*a[n-4]; od; Concatenation([1], a); # _G. C. Greubel_, May 01 2019

%K nonn

%O 0,2

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