A163215 Number of reduced words of length n in Coxeter group on 32 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 32, 992, 30752, 952816, 29521920, 914703360, 28341043200, 878114994960, 27207394552800, 842990180666400, 26119092121336800, 809270367424023600, 25074322053313752000, 776899354951763496000, 24071343043338616536000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170751, 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..665
• Index entries for linear recurrences with constant coefficients, signature (30, 30, 30, -465).

Programs

GAP

a:=[32,992,30752,952816];; for n in [5..20] do a[n]:=30*(a[n-1]+a[n-2] +a[n-3]) -465*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-31*x+495*x^4-465*x^5) )); // G. C. Greubel, Apr 28 2019

Mathematica

CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(465*t^4-30*t^3-30*t^2 - 30*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{30, 30, 30, -465}, {1, 32,992,30752,952816}, 20] (* G. C. Greubel, Dec 10 2016 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-31*x+495*x^4-465*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019

Sage

((1+x)*(1-x^4)/(1-31*x+495*x^4-465*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(465*t^4 - 30*t^3 - 30*t^2 - 30*t + 1).

From G. C. Greubel, Apr 28 2019: (Start)

a(n) = 30*(a(n-1) + a(n-2) + a(n-3)) - 465*a(n-4).

G.f.: (1+x)*(1-x^4)/(1 - 31*x + 495*x^4 - 465*x^5). (End)