A163207 Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 29, 812, 22736, 636202, 17802288, 498146166, 13939191504, 390048294510, 10914382803996, 305407698579522, 8545958486918244, 239134137088822794, 6691482951706744632, 187241958166564053774, 5239429159586654676168

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170748, 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..685
• Index entries for linear recurrences with constant coefficients, signature (27, 27, 27, -378).

Programs

GAP

a:=[29,812,22736,636202];; for n in [5..20] do a[n]:=27*(a[n-1] +a[n-2]+a[n-3] -14*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)/(1-28*x+405*x^4-378*x^5) )); // G. C. Greubel, Apr 28 2019

Mathematica

CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(378*t^4-27*t^3-27*t^2 - 27*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{27,27,27,-378}, {1,29, 812,22736,636202}, 20] (* G. C. Greubel, Dec 10 2016 *)
coxG[{4, 378, -27}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-28*x+405*x^4-378*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019

Sage

((1+x)*(1-x^4)/(1-28*x+405*x^4-378*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)/(378*t^4 - 27*t^3 - 27*t^2 - 27*t + 1).

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

a(n) = 27*(a(n-1) + a(n-2) + a(n-3) -14*a(n-4)).

G.f.: (1+x)*(1-x^4)/(1 - 28*x + 405*x^4 - 378*x^5). (End)