A164373 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 8, 56, 392, 2744, 19208, 134456, 941164, 6587952, 46114320, 322790832, 2259469968, 15815828784, 110707574544, 774930433956, 5424354927432, 37969377752376, 265777897314888, 1860391054122552, 13022357800350024

Offset

0,2

Comments

The initial terms coincide with those of A003950, 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 (6, 6, 6, 6, 6, 6, -21).

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = -21*a(n-7) + 6*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 17 2017 *)
coxG[{7, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)) \\ G. C. Greubel, Sep 17 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8) )); // G. C. Greubel, Aug 28 2019
(Sage)
def A164373_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)).list()
A164373_list(30) # G. C. Greubel, Aug 28 2019
(GAP) a:=[8, 56, 392, 2744, 19208, 134456, 941164];; for n in [8..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -21*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

Crossrefs

Sequence in context: A234274 A163347 A163924 * A164769 A165215 A165786

Adjacent sequences: A164370 A164371 A164372 * A164374 A164375 A164376

Keyword

nonn

Author

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

Status

approved