A164354 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 5, 20, 80, 320, 1280, 5120, 20470, 81840, 327210, 1308240, 5230560, 20912640, 83612160, 334295130, 1336566780, 5343813270, 21365442180, 85422543120, 341533342080, 1365506334720, 5459518355670, 21828050092440, 87272125451010

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,-6).

Formula

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

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

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

Mathematica

coxG[{7, 6, -3, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 01 2017 *)
CoefficientList[Series[(1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 15 2017 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8)) \\ G. C. Greubel, Sep 15 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8) )); // G. C. Greubel, Aug 28 2019
(Sage)
def A164354_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8)).list()
A164354_list(30) # G. C. Greubel, Aug 28 2019
(GAP) a:=[5, 20, 80, 320, 1280, 5120, 20470];; for n in [8..30] do a[n]:=3*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -6*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

Crossrefs

Sequence in context: A162925 A163316 A163878 * A164706 A165185 A165757

Adjacent sequences: A164351 A164352 A164353 * A164355 A164356 A164357

Keyword

nonn

Author

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

Status

approved