A166327 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3069, 6132, 12255, 24492, 48948, 97824, 195504, 390720, 780864, 1560576, 3118848, 6233094, 12456993, 24895608, 49754487, 99435570, 198724440, 397155696, 793725456, 1586279904, 3170219520

Offset

0,2

Comments

The initial terms coincide with those of A003945, 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,-1,2,-1,2,-1,2,-1,2,-1).

Formula

G.f.: (t^10 + t^9 + t^8 + t^7 + t^6 + t^5 + t^4 + t^3 + t^2 + t + 1) / (t^10 - 2*t^9 + t^8 - 2*t^7 + t^6 - 2*t^5 + t^4 - 2*t^3 + t^2 - 2*t + 1).

Maple

seq(coeff(series((1+t)*(1-t^11)/(1-2*t+2*t^11-t^12), t, n+1), t, n), n = 0..30); # G. C. Greubel, Mar 12 2020

Mathematica

CoefficientList[Series[(1+t)*(1-t^11)/(1-2*t+2*t^11-t^12), {t, 0, 30}], t] (* G. C. Greubel, May 09 2016 *)

Prog

(SageMath)
def A166327_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^11)/(1-2*t+2*t^11-t^12) ).list()
A166327_list(30) # G. C. Greubel, Aug 10 2019
(PARI) Vec((x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)/(x^10-2*x^9+x^8-2*x^7+x^6-2*x^5+x^4-2*x^3+x^2-2*x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 08 2026
(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; -1, 2, -1, 2, -1, 2, -1, 2, -1, 2]^(n-1)*[3; 6; 12; 24; 48; 96; 192; 384; 768; 1536])[1, 1], 1) \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A003945, A154638.

Sequence in context: A165183 A046944 A165745 * A166467 A166857 A364496

Adjacent sequences: A166324 A166325 A166326 * A166328 A166329 A166330

Keyword

nonn,easy

Author

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

Status

approved