A167959 - OEIS

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

%S 1,43,1806,75852,3185784,133802928,5619722976,236028364992,

%T 9913191329664,416354035845888,17486869505527296,734448519232146432,

%U 30846837807750150144,1295567187925506306048,54413821892871264854016

%N Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170762, 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="/A167959/b167959.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 (41,41,41,41,41,41,41,41,41,41, 41,41,41,41,41,-861).

%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)/( 861*t^16 - 41*t^15 - 41*t^14 - 41*t^13 - 41*t^12 - 41*t^11 - 41*t^10 - 41*t^9 - 41*t^8 - 41*t^7 - 41*t^6 - 41*t^5 - 41*t^4 - 41*t^3 - 41*t^2 - 41*t + 1).

%F From _G. C. Greubel_, Apr 27 2023: (Start)

%F G.f.: (1 + x)*(1 + x^16)/(1 - 42*x + 861*x^16 - 820*x^17).

%F a(n) = 41*Sum_{k=1..15} a(n-k) - 861*a(n-16). (End)

%t CoefficientList[Series[(1+x)*(1+x^16)/(1-42*x+861*x^16-820*x^17), {x, 0, 50}], x] (* _G. C. Greubel_, Jul 02 2016; Apr 27 2023 *)

%t coxG[{16, 861, -41, 40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 27 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1+x^16)/(1-42*x+861*x^16-820*x^17) )); // _G. C. Greubel_, Apr 27 2023

%o (SageMath)

%o def A167959_list(prec):

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

%o     return P( (1+x)*(1+x^16)/(1-42*x+861*x^16-820*x^17) ).list()

%o A167959_list(40) # _G. C. Greubel_, Apr 27 2023

%Y Cf. A154638, A169452, A170762.

%K nonn

%O 0,2

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