A166859 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886070, 335544240, 1342176810, 5368706640, 21474824160, 85899287040, 343597109760, 1374388285440, 5497552527360, 21990207651840, 87960820776960

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..1000

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

Formula

G.f.: (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)/(6*t^13 - 3*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).

Mathematica

CoefficientList[Series[(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)/(6*t^13 - 3*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), {t, 0, 50}], t] (* G. C. Greubel, May 25 2016 *)
coxG[{13, 6, -3, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 16 2026 *)

Prog

(PARI) Vec((1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12)*(1+x)/(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^11-3*x^12+6*x^13)+O(x^99)) \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A003947, A154638.

Sequence in context: A165757 A166331 A166495 * A167106 A167650 A269696

Adjacent sequences: A166856 A166857 A166858 * A166860 A166861 A166862

Keyword

nonn,easy

Author

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

Status

approved