A162858 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

1, 38, 1406, 51319, 1872792, 68331600, 2493179658, 90967125816, 3319062151464, 121100596329852, 4418523599533920, 161215975658220768, 5882188976123487336, 214619841546851901024, 7830703259038738949472

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170757, 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..638
• Index entries for linear recurrences with constant coefficients, signature (36, 36, -666).

Programs

GAP

a:=[38,1406,51319];; for n in [4..20] do a[n]:=36*a[n-1]+36*a[n-2]-666*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 25 2018

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series((x^3+2*x^2+2*x+1)/(666*x^3-36*x^2-36*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 25 2018

Mathematica

CoefficientList[Series[(t^3+2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 666, -36}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 27 2019 *)

PARI

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1)) \\ G. C. Greubel, Oct 24 2018

Sage

((1+x)*(1-x^3)/(1 -37*x +702*x^3 -666*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

Formula

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(666*t^3 - 36*t^2 - 36*t + 1).

a(n) = 36*a(n-1) + 36*a(n-2) - 666*a(n-3), n > 0. - Muniru A Asiru, Oct 25 2018

G.f.: (1+x)*(1-x^3)/(1 - 37*x + 702*x^3 - 666*x^4). - G. C. Greubel, Apr 27 2019