A168681 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186878, 516560616, 1549681800, 4649045256, 13947135336, 41841404712, 125524210248, 376572619080, 1129717822248

Offset

0,2

Comments

The initial terms coincide with those of A003946, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 172186878, A003946(17) = 172186884. - Klaus Brockhaus, Mar 30 2011

Computed with MAGMA using commands similar to those used to compute A154638.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

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

Formula

G.f.: (t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(3*t^17 - 2*t^16 - 2*t^15 - 2*t^14 - 2*t^13 - 2*t^12 - 2*t^11 - 2*t^10 - 2*t^9 - 2*t^8 - 2*t^7 - 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3 - 2*t^2 - 2*t + 1).

G.f.: (1+t)*(1-t^17)/(1 -3*t +5*t^17 -3*t^18). - G. C. Greubel, Feb 22 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -3*t +5*t^17 -3*t^18), {t, 0, 40}], t] (* G. C. Greubel, Aug 03 2016, Feb 22 2021 *)
coxG[{17, 3, -2, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Feb 22 2021 *)

Prog

(Magma)
R<t>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -3*t +5*t^17 -3*t^18) )); // G. C. Greubel, Feb 22 2021
(Sage)
def A168681_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -3*t +5*t^17 -3*t^18) ).list()
A168681_list(40) # G. C. Greubel, Feb 22 2021

Crossrefs

Cf. A003946 (G.f.: (1+x)/(1-3*x)).

Sequence in context: A167105 A167649 A167882 * A168729 A168777 A168825

Adjacent sequences: A168678 A168679 A168680 * A168682 A168683 A168684

Keyword

nonn,easy

Author

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

Status

approved