%I #32 Sep 08 2022 08:45:47

%S 1,3,6,12,24,48,93,180,351,684,1332,2592,5046,9825,19128,37239,72498,

%T 141144,274788,534972,1041513,2027676,3947595,7685400,14962368,

%U 29129580,56711106,110408373,214949232,418475259,814711182,1586125572,3087958512

%N Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%C Also, coordination sequence for (6,6,6) tiling of hyperbolic plane. - _N. J. A. Sloane_, Dec 29 2015

%C The initial terms coincide with those of A003945, 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="/A163876/b163876.txt">Table of n, a(n) for n = 0..1000</a>

%H J. W. Cannon, P. Wagreich, <a href="http://dx.doi.org/10.1007/BF01444714">Growth functions of surface groups</a>, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, 1, 1, -1).

%F G.f.: (x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/(x^6 - x^5 - x^4 - x^3 - x^2 - x + 1).

%F G.f.: (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7). - _G. C. Greubel_, Apr 25 2019

%F a(n) = -a(n-6) + Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 07 2021

%t coxG[{6,1,-1,40}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Mar 22 2015 *)

%t CoefficientList[Series[(1+x)*(1-x^6)/(1-2*x+2*x^6-x^7), {x,0,40}], x] (* _G. C. Greubel_, Aug 06 2017, modified Apr 25 2019 *)

%o (PARI) x='x+O('x^40); Vec((x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/(x^6-x^5- x^4-x^3-x^2-x+1)) \\ _G. C. Greubel_, Aug 06 2017

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7) )); // _G. C. Greubel_, Apr 25 2019

%o (Sage) ((1+x)*(1-x^6)/(1-2*x+2*x^6-x^7)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 25 2019

%Y Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

%K nonn,easy

%O 0,2

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