A163315 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 4, 12, 36, 108, 318, 936, 2760, 8136, 23976, 70662, 208260, 613788, 1808964, 5331420, 15712878, 46309320, 136483800, 402247944, 1185513624, 3493970742, 10297504260, 30349021740, 89445276900, 263615006412, 776931706398

Offset

0,2

Comments

The initial terms coincide with those of A003946, 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..1000

Index entries for linear recurrences with constant coefficients, signature (2, 2, 2, 2, -3).

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(3*t^5 - 2*t^4 - 2*t^3 - 2*t^2 - 2*t + 1).

a(n) = 2*a(n-1)+2*a(n-2)+2*a(n-3)+2*a(n-4)-3*a(n-5). - Wesley Ivan Hurt, May 10 2021

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6), {x, 0, 30}], x] (* or *) Join[{1}, LinearRecurrence[{2, 2, 2, 2, -3}, {1, 4, 12, 36, 108, 318}, 30]] (* G. C. Greubel, Dec 18 2016 *)
coxG[{4, 3, -2}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6)) \\ G. C. Greubel, Dec 18 2016
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6) )); // G. C. Greubel, May 12 2019
(Sage) ((1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Crossrefs

Sequence in context: A003212 A156945 A006817 * A326339 A334877 A003119

Adjacent sequences: A163312 A163313 A163314 * A163316 A163317 A163318

Keyword

nonn,easy

Author

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

Status

approved