|
%I #24 Sep 08 2022 08:45:46
%S 1,34,1122,37026,1221297,40284288,1328771136,43829305344,
%T 1445702699760,47686274735616,1572924224543232,51882656590093824,
%U 1711341215834452224,56448319139710451712,1861938872397761101824,61415759005426222645248
%N Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C The initial terms coincide with those of A170753, 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="/A163217/b163217.txt">Table of n, a(n) for n = 0..650</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (32, 32, 32, -528).
%F G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(528*t^4 - 32*t^3 - 32*t^2 - 32*t + 1).
%F From _G. C. Greubel_, Apr 28 2019: (Start)
%F a(n) = 32*(a(n-1) + a(n-2) + a(n-3)) - 528*a(n-4).
%F G.f.: (1+x)*(1-x^4)/(1 - 33*x + 560*x^4 - 528*x^5).
%t CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(528*t^4-32*t^3-32*t^2 - 32*t+1), {t,0,20}], t] (* or *)
%t LinearRecurrence[{32, 32, 32, -528}, {1, 34, 1122, 37026, 1221297}, 20] (* _G. C. Greubel_, Dec 11 2016; simplified by _Georg Fischer_, Apr 08 2019 *)
%t coxG[{4,528,-32}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jul 06 2018 *)
%o (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-33*x+560*x^4-528*x^5)) \\ _G. C. Greubel_, Dec 11 2016, modified Apr 28 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-33*x+560*x^4-528*x^5) )); // _G. C. Greubel_, Apr 28 2019
%o (Sage) ((1+x)*(1-x^4)/(1-33*x+560*x^4-528*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 28 2019
%o (GAP) a:=[34,1122,37026,1221297];; for n in [5..20] do a[n]:=32*(a[n-1]+ a[n-2]+a[n-3]) -528*a[n-4]; od; Concatenation([1], a); # _G. C. Greubel_, Apr 28 2019
%Y Cf. A154638, A170753.
%K nonn,easy
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
|
