A167958 - OEIS

%I #21 Dec 03 2025 17:11:32

%S 1,42,1722,70602,2894682,118681962,4865960442,199504378122,

%T 8179679503002,335366859623082,13750041244546362,563751691026400842,

%U 23113819332082434522,947666592615379815402,38854330297230572431482

%N Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170761, 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="/A167958/b167958.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,-820).

%F G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*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)/( 820*t^16 - 40*t^15 - 40*t^14 - 40*t^13 - 40*t^12 - 40*t^11 - 40*t^10 - 40*t^9 - 40*t^8 - 40*t^7 - 40*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).

%F From _G. C. Greubel_, Jul 14 2023: (Start)

%F G.f.: (1 + t)*(1 - t^16)/(1 - 41*t + 860*t^16 - 820*t^17).

%F a(n) = -820*a(n-16) + 40*Sum_{j=1..15} a(n-j). (End)

%t CoefficientList[Series[(1+t)*(1-t^16)/(1-41*t+860*t^16 -820*t^17), {t, 0, 40}], t] (* _G. C. Greubel_, Jul 02 2016; Jul 14 2023 *)

%t coxG[{16, 820, -40, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Jul 14 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) )); // _G. C. Greubel_, Jul 14 2023

%o (SageMath)

%o def A167958_list(prec):

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

%o     return P( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) ).list()

%o A167958_list(40) # _G. C. Greubel_, Jul 14 2023

%Y Cf. A154638, A169452, A170761.

%K nonn

%O 0,2

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