A163347 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 8, 56, 392, 2744, 19180, 134064, 937104, 6550320, 45786384, 320044452, 2237094216, 15637173048, 109303031880, 764022547512, 5340478146444, 37329666414768, 260932440209616, 1823904280240560, 12748996716570576

Offset: 0

Author

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

Keywords

• nonn

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, -21).

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6) )); // G. C. Greubel, May 12 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{6,6,6,6,-21}, {1,8,56,392,2744,19180}, 30] (* G. C. Greubel, Dec 19 2016 *)
coxG[{5, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6)) \\ G. C. Greubel, Dec 19 2016

Sage

((1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

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

a(n) = 6*a(n-1)+6*a(n-2)+6*a(n-3)+6*a(n-4)-21*a(n-5). - Wesley Ivan Hurt, May 10 2021