A163922 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 6, 30, 150, 750, 3750, 18735, 93600, 467640, 2336400, 11673000, 58320000, 291375210, 1455753000, 7273154040, 36337737000, 181548627000, 907043385000, 4531720872060, 22641137570400, 113118421225440, 565156109349600

Offset

0,2

Comments

The initial terms coincide with those of A003948, 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 (4,4,4,4,4,-10).

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

a(n) = -10*a(n-6) + 4*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

coxG[{6, 10, -4, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 13 2017 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7), {t, 0, 30}], t] (* G. C. Greubel, Aug 07 2017 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7)) \\ G. C. Greubel, Aug 07 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7) )); // G. C. Greubel, Aug 10 2019
(Sage)
def A163922_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7)).list()
A163922_list(30) # G. C. Greubel, Aug 10 2019
(GAP) a:=[6, 30, 150, 750, 3750, 18735];; for n in [7..30] do a[n]:=4*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -10*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Crossrefs

Sequence in context: A006819 A163317 A342807 * A164365 A164741 A165213

Adjacent sequences: A163919 A163920 A163921 * A163923 A163924 A163925

Keyword

nonn

Author

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

Status

approved