A162871 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

1, 39, 1482, 55575, 2083692, 78111033, 2928135600, 109766289945, 4114781688966, 154249795892907, 5782323668697966, 216760526662519203, 8125647855742321632, 304604136609884440797, 11418619374984439210164

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170758, 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..615
• Index entries for linear recurrences with constant coefficients, signature (37, 37, -703).

Programs

GAP

a:=[39,1482,55575];; for n in [4..15] do a[n]:=37*a[n-1]+37*a[n-2]-703*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series((x^3+2*x^2+2*x+1)/(703*x^3-37*x^2-37*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 24 2018

Mathematica

coxG[{3,703,-37}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 25 2018 *)
CoefficientList[Series[(t^3+2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)

PARI

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1)) \\ G. C. Greubel, Oct 24 2018

Sage

((1+x)*(1-x^3)/(1 -38*x +740*x^3 -703*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 27 2019

Formula

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(703*t^3 - 37*t^2 - 37*t + 1).

a(n) = 37*a(n-1) + 37*a(n-2) - 703*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018

G.f.: (1+x)*(1-x^3)/(1 - 38*x + 740*x^3 - 703*x^4). - G. C. Greubel, Apr 27 2019