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Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
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%I #16 Dec 03 2024 03:23:20

%S 1,13,156,1872,22464,269568,3234816,38817792,465813504,5589762048,

%T 67077144576,804925734912,9659108818866,115909305825456,

%U 1390911669894318,16690940038597968,200291280461569440,2403495365519559168,28841944386003420672,346103332629265575936

%N Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%C The initial terms coincide with those of A170732, 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="/A166558/b166558.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (11,11,11,11,11,11,11,11,11,11,11,-66).

%F 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)/(66*t^12 - 11*t^11 - 11*t^10 - 11*t^9 - 11*t^8 - 11*t^7 - 11*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t +1).

%F From _G. C. Greubel_, Dec 03 2024: (Start)

%F a(n) = 11*Sum_{j=1..11} a(n-j) - 66*a(n-12).

%F G.f.: (1+x)*(1-x^12)/(1 - 12*x + 77*x^12 - 66*x^13). (End)

%t CoefficientList[Series[(1+t)*(1-t^12)/(1-12*t+77*t^12-66*t^13), {t,0,50}], t] (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)

%t coxG[{12,66,-11}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Dec 03 2024 *)

%o (Magma)

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

%o Coefficients(R!( (1+x)*(1-x^12)/(1-12*x+77*x^12-66*x^13) )); // _G. C. Greubel_, Dec 03 2024

%o (SageMath)

%o def A166558_list(prec):

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

%o return P( (1+x)*(1-x^12)/(1-12*x+77*x^12-66*x^13) ).list()

%o A166558_list(40) # _G. C. Greubel_, Dec 03 2024

%Y Cf. A154638, A169452, A170732.

%K nonn

%O 0,2

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