A163221 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 38, 1406, 52022, 1924111, 71166096, 2632183848, 97355219328, 3600827035866, 133181923185576, 4925930761424952, 182192847843197736, 6738672428195210748, 249239784283952410080, 9218502714272560450272

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..635
• Index entries for linear recurrences with constant coefficients, signature (36, 36, 36, -666).

Programs

GAP

a:=[38,1406,52022,1924111];; for n in [5..20] do a[n]:=36*(a[n-1]+ a[n-2]+a[n-3]) -666*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, May 01 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-37*x+702*x^4-666*x^5) )); // G. C. Greubel, May 01 2019

Mathematica

coxG[{4,666,-36}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 09 2015 *)
CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(666*t^4-36*t^3-36*t^2 - 36*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{36, 36, 36, -666}, {1, 38, 1406, 52022, 1924111}, 20] (* G. C. Greubel, Dec 11 2016; modified by Georg Fischer, Apr 08 2019 *)

PARI

my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(666*t^4-36*t^3 - 36*t^2-36*t+1)) \\ G. C. Greubel, Dec 11 2016

Sage

((1+x)*(1-x^4)/(1-37*x+702*x^4-666*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(666*t^4 - 36*t^3 - 36*t^2 - 36*t + 1).

a(n) = 36*a(n-1)+36*a(n-2)+36*a(n-3)-666*a(n-4). - Wesley Ivan Hurt, May 06 2021