A164697 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8742, 26208, 78576, 235584, 706320, 2117664, 6349104, 19035648, 57071982, 171111132, 513019140, 1538115228, 4611520836, 13826093148, 41452886916, 124282529820, 372619336494, 1117173669768

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 (2,2,2,2,2,2,2,-3).

Formula

G.f.: (x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/( 3*x^8 - 2*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-8) + 2*Sum_{k=1..7} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 16 2019

Mathematica

CoefficientList[Series[(x^8 +2x^7 +2x^6 +2x^5 +2x^4 +2x^3 +2x^2 +2x +1)/( 3x^8 -2x^7 -2x^6 -2x^5 -2x^4 -2x^3 -2x^2 -2x +1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
coxG[{8, 3, -2, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 03 2019 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9)) \\ G. C. Greubel, Sep 16 2019
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9) )); // G. C. Greubel, Sep 16 2019
(SageMath)
def A164697_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9)).list()
A164697_list(30) # G. C. Greubel, Sep 16 2019
(GAP) a:=[4, 12, 36, 108, 324, 972, 2916, 8742];; for n in [9..30] do a[n]:=2*Sum([1..7], j-> a[n-j]) -3*a[n-8]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019

Crossrefs

Sequence in context: A336262 A164353 A347506 * A165184 A165756 A166328

Adjacent sequences: A164694 A164695 A164696 * A164698 A164699 A164700

Keyword

nonn,easy

Author

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

Status

approved