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A166690 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166690 #19 Sep 08 2022 08:45:48
%S A166690 1,38,1406,52022,1924814,71218118,2635070366,97497603542,
%T A166690 3607411331054,133474219248998,4938546112212926,182726206151878262,
%U A166690 6760869627619494991,250152176221921288656,9255630520211086718568
%N A166690 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166690 The initial terms coincide with those of A170757, although the two sequences are eventually different.
%C A166690 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166690 G. C. Greubel, <a href="/A166690/b166690.txt">Table of n, a(n) for n = 0..500</a>
%H A166690 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, -666).
%F A166690 G.f.: (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)/(666*t^12 - 36*t^11 - 36*t^10 - 36*t^9 -36*t^8 -36*t^7 -36*t^6 - 36*t^5 - 36*t^4 - 36*t^3 - 36*t^2 - 36*t + 1).
%F A166690 G.f.: (1+x)*(1-x^12)/(1 -37*x +702*x^6 -666*x^7). - _G. C. Greubel_, Apr 26 2019
%F A166690 a(n) = -666*a(n-12) + 36*Sum_{k=1..11} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%t A166690 CoefficientList[Series[(1+x)*(1-x^12)/(1-37*x+702*x^6-666*x^7), {x, 0, 20}], x] (* _G. C. Greubel_, May 23 2016, modified Apr 26 2019 *)
%t A166690 coxG[{12, 666, -36}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 26 2019 *)
%o A166690 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^12)/(1-37*x+702*x^6-666*x^7)) \\ _G. C. Greubel_, Apr 26 2019
%o A166690 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^12)/(1-37*x+702*x^6-666*x^7) )); // _G. C. Greubel_, Apr 26 2019
%o A166690 (Sage) ((1+x)*(1-x^12)/(1-37*x+702*x^6-666*x^7)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%K A166690 nonn
%O A166690 0,2
%A A166690 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009