A169401 - OEIS

%I #22 May 08 2018 00:48:49

%S 1,4,12,36,108,324,972,2916,8748,26244,78732,236196,708588,2125764,

%T 6377292,19131876,57395628,172186884,516560652,1549681956,4649045868,

%U 13947137604,41841412812,125524238436,376572715308,1129718145924

%N Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^32 = I.

%C The initial terms coincide with those of A003946, although the two sequences are eventually different.

%C First disagreement is at index 32, the difference is 6. - _Klaus Brockhaus_, Jun 26 2011

%C Computed with Magma using commands similar to those used to compute A154638.

%H Vincenzo Librandi, <a href="/A169401/b169401.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -3).

%F G.f.: (t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*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)/(3*t^32 - 2*t^31 - 2*t^30 - 2*t^29 - 2*t^28 - 2*t^27 - 2*t^26 - 2*t^25 - 2*t^24 - 2*t^23 - 2*t^22 - 2*t^21 - 2*t^20 - 2*t^19 - 2*t^18 - 2*t^17 - 2*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).

%F G.f.: (1+2*sum(k=1..31,x^k)+x^32)/(1-2*sum(k=1..31,x^k)+3*x^32).

%t With[{num=Total[2t^Range[31]]+t^32+1,den=Total[-2 t^Range[31]]+3t^32+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* _Harvey P. Dale_, Nov 30 2011 *)

%o (PARI) x='x+O('x^66); /* that many terms */

%o Vec((1+2*sum(k=1,31,x^k)+x^32)/(1-2*sum(k=1,31,x^k)+3*x^32))  /* show terms */

%o /* _Joerg Arndt_, Jun 26 2011 */

%Y Cf. A003946 (G.f.: (1+x)/(1-3*x) ).

%K nonn

%O 0,2

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