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A163222 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%I A163222 #14 Sep 08 2022 08:45:46
%S A163222 1,39,1482,56316,2139267,81263988,3086962281,117263934684,
%T A163222 4454486050560,169211838474861,6427822638540342,244172655087350379,
%U A163222 9275347010187982854,352341101130365494992,13384324210123816783899
%N A163222 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C A163222 The initial terms coincide with those of A170758, although the two sequences are eventually different.
%C A163222 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163222 G. C. Greubel, <a href="/A163222/b163222.txt">Table of n, a(n) for n = 0..630</a>
%H A163222 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (37, 37, 37, -703).
%F A163222 G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(703*t^4 - 37*t^3 - 37*t^2 - 37*t + 1).
%F A163222 a(n) = 37*a(n-1)+37*a(n-2)+37*a(n-3)-703*a(n-4). - _Wesley Ivan Hurt_, May 06 2021
%t A163222 CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(703*t^4-37*t^3-37*t^2 - 37*t+1), {t,0,20}], t] (* or *) Join[{1}, LinearRecurrence[{37, 37, 37, -703}, {39, 1482, 56316, 2139267}, 20]] (* _G. C. Greubel_, Dec 11 2016 *)
%t A163222 coxG[{4, 703, -37}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 30 2019 *)
%o A163222 (PARI) my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(703*t^4-37*t^3 - 37*t^2-37*t+1)) \\ _G. c. Greubel_, Dec 11 2016
%o A163222 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-38*x+740*x^4-703*x^5) )); // _G. C. Greubel_, Apr 30 2019
%o A163222 (Sage) ((1+x)*(1-x^4)/(1-38*x+740*x^4-703*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 30 2019
%K A163222 nonn
%O A163222 0,2
%A A163222 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009