A162885 Number of reduced words of length n in Coxeter group on 45 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

1, 45, 1980, 86130, 3746160, 162915390, 7084967670, 308115104220, 13399485132330, 582724430755830, 25341851494598760, 1102080851855063190, 47927918932540448670, 2084316599215116583020, 90643945794494362584930

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170764, 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..609
• Index entries for linear recurrences with constant coefficients, signature (43, 43, -946).

Programs

GAP

a:=[45,1980,86130];; for n in [4..20] do a[n]:=43*a[n-1]+43*a[n-2] -946*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)/(946*t^3-43*t^2-43*t+1))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series((x^3+2*x^2+2*x+1)/(946*x^3-43*x^2-43*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 24 2018

Mathematica

CoefficientList[Series[(t^3+2*t^2+2*t+1)/(946*t^3-43*t^2-43*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 946, -43}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)

PARI

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(946*t^3-43*t^2-43*t+1)) \\ G. C. Greubel, Oct 24 2018

Sage

((1+x)*(1-x^3)/(1-44*x+990*x^3-946*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(946*t^3 - 43*t^2 - 43*t + 1).

a(n) = 43*a(n-1) + 43*a(n-2) - 946*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018

G.f.: (1+x)*(1-x^3)/(1 - 44*x + 990*x^3 - 946*x^4). - G. C. Greubel, Apr 28 2019