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A164113 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164113 #17 Sep 08 2022 08:45:47
%S A164113 1,43,1806,75852,3185784,133802928,5619722073,236028289140,
%T A164113 9913186551891,416353768315884,17486855460998532,734447811414657312,
%U A164113 30846803125630618266,1295565523217549867745,54413743236663181589148
%N A164113 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164113 The initial terms coincide with those of A170762, although the two sequences are eventually different.
%C A164113 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164113 G. C. Greubel, <a href="/A164113/b164113.txt">Table of n, a(n) for n = 0..614</a>
%H A164113 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (41,41,41,41,41,-861).
%F A164113 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(861*t^6 - 41*t^5 - 41*t^4 - 41*t^3 - 41*t^2 - 41*t + 1).
%F A164113 a(n) = -861*a(n-6) + 41*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164113 seq(coeff(series((1+t)*(1-t^6)/(1-42*t+902*t^6-861*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 16 2019
%t A164113 CoefficientList[Series[(1+t)*(1-t^6)/(1-42*t+902*t^6-861*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 11 2017 *)
%t A164113 coxG[{6, 861, -41}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 16 2019 *)
%o A164113 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-42*t+902*t^6-861*t^7)) \\ _G. C. Greubel_, Sep 11 2017
%o A164113 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-42*t+902*t^6-861*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A164113 (Sage)
%o A164113 def A164113_list(prec):
%o A164113     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164113     return P((1+t)*(1-t^6)/(1-42*t+902*t^6-861*t^7)).list()
%o A164113 A164113_list(30) # _G. C. Greubel_, Aug 10 2019
%o A164113 (GAP) a:=[43, 1806, 75852, 3185784, 133802928, 5619722073];; for n in [7..30] do a[n]:=41*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -861*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A164113 nonn
%O A164113 0,2
%A A164113 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009