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A163214 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%I A163214 #23 Sep 08 2022 08:45:46
%S A163214 1,31,930,27900,836535,25082100,752044965,22548807900,676088221260,
%T A163214 20271372436125,607803134933490,18223958540698875,546414860017738110,
%U A163214 16383333982098029400,491226816855341457015,14728612983261055500600
%N A163214 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C A163214 The initial terms coincide with those of A170750, although the two sequences are eventually different.
%C A163214 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163214 G. C. Greubel, <a href="/A163214/b163214.txt">Table of n, a(n) for n = 0..670</a>
%H A163214 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (29,29,29,-435).
%F A163214 G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(435*t^4 - 29*t^3 - 29*t^2 - 29*t + 1).
%F A163214 a(n) = 29*(a(n-1) + a(n-2) + a(n-3) - 15*a(n-4)). - _G. C. Greubel_, Apr 28 2019
%t A163214 coxG[{4,435,-29}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Apr 24 2016 *)
%t A163214 CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(435*t^4-29*t^3-29*t^2 - 29*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{29,29,29,-435}, {1,31, 930,27900,836535}, 20] (* _G. C. Greubel_, Dec 10 2016 *)
%o A163214 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5)) \\ _G. C. Greubel_, Dec 10 2016, modified Apr 28 2019
%o A163214 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5) )); // _G. C. Greubel_, Apr 28 2019
%o A163214 (Sage) ((1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 28 2019
%o A163214 (GAP) a:=[31,930,27900,836535];; for n in [5..20] do a[n]:=29*(a[n-1]+ a[n-2] +a[n-3] -15*a[n-4]); od; Concatenation([1], a); # _G. C. Greubel_, Apr 28 2019
%K A163214 nonn
%O A163214 0,2
%A A163214 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009