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%I #19 Sep 08 2022 08:45:46
%S 1,39,1482,55575,2083692,78111033,2928135600,109766289945,
%T 4114781688966,154249795892907,5782323668697966,216760526662519203,
%U 8125647855742321632,304604136609884440797,11418619374984439210164
%N Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.
%C The initial terms coincide with those of A170758, 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="/A162871/b162871.txt">Table of n, a(n) for n = 0..615</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (37, 37, -703).
%F G.f.: (t^3 + 2*t^2 + 2*t + 1)/(703*t^3 - 37*t^2 - 37*t + 1).
%F a(n) = 37*a(n-1) + 37*a(n-2) - 703*a(n-3), n > 0. - _Muniru A Asiru_, Oct 24 2018
%F G.f.: (1+x)*(1-x^3)/(1 - 38*x + 740*x^3 - 703*x^4). - _G. C. Greubel_, Apr 27 2019
%p seq(coeff(series((x^3+2*x^2+2*x+1)/(703*x^3-37*x^2-37*x+1),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 24 2018
%t coxG[{3,703,-37}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jun 25 2018 *)
%t CoefficientList[Series[(t^3+2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1), {t, 0, 20}], t] (* _G. C. Greubel_, Oct 24 2018 *)
%o (PARI) my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1)) \\ _G. C. Greubel_, Oct 24 2018
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1))); // _G. C. Greubel_, Oct 24 2018
%o (GAP) a:=[39,1482,55575];; for n in [4..15] do a[n]:=37*a[n-1]+37*a[n-2]-703*a[n-3]; od; Concatenation([1],a); # _Muniru A Asiru_, Oct 24 2018
%o (Sage) ((1+x)*(1-x^3)/(1 -38*x +740*x^3 -703*x^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 27 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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