A163265 Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 47, 2162, 99452, 4573711, 210340980, 9673398765, 444871172700, 20459237269140, 940902479912925, 43271284508242650, 1990008638480367675, 91518761835509986350, 4208868045065726973000, 193562170919821248573375

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170766, 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..600
• Index entries for linear recurrences with constant coefficients, signature (45, 45, 45, -1035).

Programs

GAP

a:=[47,2162,99452,4573711];; for n in [5..20] do a[n]:=45*(a[n-1]+a[n-2] +a[n-3]-23*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)/(1-46*x+1080*x^4-1035*x^5) )); // G. C. Greubel, May 01 2019

Mathematica

CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(1035*t^4-45*t^3-45*t^2 - 45*t+1), {t,0,20}], t] (* or *) LinearRecurrence[ {45, 45, 45, -1035}, {1,47,2162,99452,4573711}, 20] (* G. C. Greubel, Dec 12 2016 *)
coxG[{4, 1035, -45}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 01 2019 *)

PARI

my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(1035*t^4-45*t^3- 45*t^2-45*t+1)) \\ G. C. Greubel, Dec 12 2016

Sage

((1+x)*(1-x^4)/(1-46*x+1080*x^4-1035*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)/(1035*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).

a(n) = 45*a(n-1)+45*a(n-2)+45*a(n-3)-1035*a(n-4). - Wesley Ivan Hurt, May 10 2021