A162879 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

1, 42, 1722, 69741, 2824080, 114340800, 4629407580, 187434189600, 7588784431200, 307252630616400, 12439960566432000, 503665724648352000, 20392280251485912000, 825637071380896320000, 33428168171083640640000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170761, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (40, 40, -820).

Crossrefs

Cf. A154638, A170761.

Programs

GAP

a:=[42,1722,69741];; for n in [4..20] do a[n]:=40*a[n-1]+40*a[n-2] -820*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

Magma

I:=[1,42,1722,69741]; [n le 4 select I[n] else 40*Self(n-1) +40*Self(n-2)-820*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Apr 14 2017

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 +2*t^2+2*t+1)/(820*t^3-40*t^2-40*t+1))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series((x^3+2*x^2+2*x+1)/(820*x^3-40*x^2-40*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)/(820*t^3-40*t^2-40*t+1), {t, 0, 20}], t] (* Wesley Ivan Hurt, Apr 12 2017 *)
Join[{1}, LinearRecurrence[{40, 40, -820}, {42, 1722, 69741}, 20]] (* Vincenzo Librandi, Apr 14 2017 *)
coxG[{3, 820, -40}] (* 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)/(820*t^3-40*t^2-40*t+1)) \\ G. C. Greubel, Oct 24 2018

Sage

((1+x)*(1-x^3)/(1 -41*x +860*x^3 -820*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)/(820*t^3 - 40*t^2 - 40*t + 1).

a(n) = 40*a(n-1) + 40*a(n-2) - 820*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018

G.f.: (1+x)*(1-x^3)/(1 - 41*x + 860*x^3 - 820*x^4). - G. C. Greubel, Apr 27 2019