A166328 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236190, 708552, 2125608, 6376680, 19129608, 57387528, 172158696, 516464424, 1549358280, 4647969864, 13943594664, 41829839238, 125486683524, 376451548188, 1129329137988

Offset

0,2

Comments

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,2,2,2,2,2,2,2,2,-3).

Formula

G.f.: (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^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).

Maple

seq(coeff(series((1+t)*(1-t^11)/(1-3*t+5*t^11-3*t^12), t, n+1), t, n), n = 0..30); # G. C. Greubel, Mar 12 2020

Mathematica

CoefficientList[Series[(1+t)*(1-t^11)/(1-3*t+5*t^11-3*t^12), {t, 0, 30}], t] (* G. C. Greubel, May 09 2016 *)
coxG[{11, 3, -2}] (* The coxG program is in A169452 *) (* G. C. Greubel, Mar 12 2020 *)

Prog

(SageMath)
def A166328_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^11)/(1-3*t+5*t^11-3*t^12) ).list()
A166328_list(30) # G. C. Greubel, Aug 10 2019
(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; -3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]^(n-1)*[4; 12; 36; 108; 324; 972; 2916; 8748; 26244; 78732; 236190])[1, 1], 1) \\ Charles R Greathouse IV, Jun 08 2026
(PARI) Vec((1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)*(1+x)/(1-2*x-2*x^2-2*x^3-2*x^4-2*x^5-2*x^6-2*x^7-2*x^8-2*x^9-2*x^10+3*x^11)+O(x^99)) \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A003946, A154638.

Sequence in context: A164697 A165184 A165756 * A166468 A166858 A167105

Adjacent sequences: A166325 A166326 A166327 * A166329 A166330 A166331

Keyword

nonn,easy

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Status

approved