A164610 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 13, 156, 1872, 22464, 269568, 3234816, 38817714, 465811632, 5589728430, 67076607312, 804917681568, 9658992904704, 115907683567104, 1390889427339126, 16690639822542972, 200287278204994266

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170732, 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..920
• Index entries for linear recurrences with constant coefficients, signature (11,11,11,11,11,11,-66).

Programs

GAP

a:=[13,156,1872,22464,269568,3234816,38817714];; for n in [8..20] do a[n]:=11*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -66*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Sep 15 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-12*t+77*t^7-66*t^8) )); // G. C. Greubel, Sep 15 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-12*t+77*t^7-66*t^8), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 15 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-12*t+77*t^7-66*t^8), {t, 0, 20}], t] (* G. C. Greubel, Aug 10 2017 *)
coxG[{7, 6, -11}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 15 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-12*t+77*t^7-66*t^8)) \\ G. C. Greubel, Aug 10 2017

Sage

def A164610_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-12*t+77*t^7-66*t^8)).list()
A164610_list(20) # G. C. Greubel, Sep 15 2019

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(66*t^7 - 11*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t + 1).

a(n) = -66*a(n-7) + 11*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021