A166495 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971510, 83886000, 335543850, 1342174800, 5368696800, 21474777600, 85899072000, 343596134400, 1374383923200, 5497533235200, 21990123110400, 87960453120000

Offset

0,2

Comments

The initial terms coincide with those of A003947, 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..500

Index entries for linear recurrences with constant coefficients, signature (3,3,3,3,3,3,3,3,3,3,3,-6).

Formula

G.f.: (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)/(6*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

From G. C. Greubel, Aug 02 2024: (Start)

a(n) = 3*Sum_{j=1..11} a(n-j) - 6*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 4*x + 9*x^12 - 6*x^13). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-4*t+9*t^12-6*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 15 2016; Aug 02 2024 *)
coxG[{12, 6, -3, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Feb 19 2018 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
f:= func< p, q, x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
Coefficients(R!( f(6, 3, x) )); // G. C. Greubel, Aug 02 2024
(SageMath)
def f(p, q, x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166495_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( f(6, 3, x) ).list()
A166495_list(30) # G. C. Greubel, Aug 02 2024
(PARI) Vec((1+x^4+x^8)*(1+x^2)*(1+x)^2/(1-3*x-3*x^2-3*x^3-3*x^4-3*x^5-3*x^6-3*x^7-3*x^8-3*x^9-3*x^10-3*x^11+6*x^12)+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; 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; -6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]^(n-1)*[5; 20; 80; 320; 1280; 5120; 20480; 81920; 327680; 1310720; 5242880; 20971510])[1, 1], 1) \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A003947, A154638, A169452.

Sequence in context: A165185 A165757 A166331 * A166859 A167106 A167650

Adjacent sequences: A166492 A166493 A166494 * A166496 A166497 A166498

Keyword

nonn,easy,changed

Author

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

Status

approved