A164353 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 4, 12, 36, 108, 324, 972, 2910, 8712, 26088, 78120, 233928, 700488, 2097576, 6281094, 18808452, 56321052, 168650820, 505017180, 1512250884, 4528366236, 13559985966, 40604758920, 121589096856, 364092999624, 1090259865432

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: (x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/(3*x^7 - 2*x^6 - 2*x^5 - 2*x^4 - 2*x^3 - 2*x^2 - 2*x + 1).

a(n) = -3*a(n-7) + 2*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-3*t+5*t^7-3*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 24 2019

Mathematica

CoefficientList[Series[(x^7 + 2 x^6 + 2 x^5 + 2 x^4 + 2 x^3 + 2 x^2 + 2 x + 1)/(3 x^7 - 2 x^6 - 2 x^5 - 2 x^4 - 2 x^3 - 2 x^2 - 2 x + 1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
coxG[{7, 3, -2, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 15 2015 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-3*t+5*t^7-3*t^8)) \\ G. C. Greubel, Sep 15 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-3*t+5*t^7-3*t^8) )); // G. C. Greubel, Aug 24 2019
(Sage)
def A164353_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-3*t+5*t^7-3*t^8)).list()
A164353_list(30) # G. C. Greubel, Aug 24 2019

Crossrefs

Sequence in context: A156946 A163877 A336262 * A347506 A164697 A165184

Adjacent sequences: A164350 A164351 A164352 * A164354 A164355 A164356

Keyword

nonn,easy

Author

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

Status

approved