A164352 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 3, 6, 12, 24, 48, 96, 189, 372, 735, 1452, 2868, 5664, 11184, 22086, 43617, 86136, 170103, 335922, 663384, 1310064, 2587140, 5109132, 10089609, 19925148, 39348555, 77706264, 153455784, 303047352, 598463580, 1181857074, 2333953461

Offset

0,2

Comments

The initial terms coincide with those of A003945, 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, -1, 2, -1, 2, -1).

Formula

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

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1-t^7)/((1-t)*(1-2*t+t^2-2*t^3+t^4-2*t^5+t^6)), t, n+1), t, n), n = 0 .. 35); # G. C. Greubel, Aug 24 2019

Mathematica

CoefficientList[Series[(1-t^7)/((1-t)*(1-2*t+t^2-2*t^3+t^4-2*t^5+t^6)), {t, 0, 35}], t] (* G. C. Greubel, Sep 15 2017 *)
LinearRecurrence[{2, -1, 2, -1, 2, -1}, {1, 3, 6, 12, 24, 48, 96}, 40] (* Harvey P. Dale, May 21 2021 *)

Prog

(PARI) (t='t+O('t^35)); Vec((1-t^7)/((1-t)*(1-2*t+t^2-2*t^3+t^4-2*t^5 +t^6))) \\ G. C. Greubel, Sep 15 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-t^7)/( (1-t)*(1-2*t+t^2-2*t^3+t^4-2*t^5+t^6)) )); // G. C. Greubel, Aug 24 2019
(Sage)
def A164352_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1-t^7)/((1-t)*(1-2*t+t^2-2*t^3+t^4-2*t^5+t^6))).list()
A164352_list(35) # G. C. Greubel, Aug 24 2019
(GAP) a:=[3, 6, 12, 24, 48, 96];; for n in [7..35] do a[n]:=2*a[n-1] -a[n-2]+2*a[n-3]-a[n-4]+2*a[n-5]-a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 24 2019

Crossrefs

Sequence in context: A163876 A033893 A006851 * A115829 A115805 A356040

Adjacent sequences: A164349 A164350 A164351 * A164353 A164354 A164355

Keyword

nonn

Author

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

Status

approved