A167950 Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 34, 1122, 37026, 1221858, 40321314, 1330603362, 43909910946, 1449027061218, 47817893020194, 1577990469666402, 52073685498991266, 1718431621466711778, 56708243508401488674, 1871372035777249126242, 61755277180649221165986

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170753, 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..500
• Index entries for linear recurrences with constant coefficients, signature (32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,-528).

Crossrefs

Cf. A154638, A169452, A170753.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-33*x+560*x^16-528*x^17) )); // G. C. Greubel, Sep 06 2023

Mathematica

coxG[{16,528,-32}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 24 2015 *)
CoefficientList[Series[(1+t)*(1-t^16)/(1-33*t+560*t^16-528*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 06 2023 *)

SageMath

def A167955_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-33*x+560*x^16-528*x^17) ).list()
A167955_list(40) # G. C. Greubel, Sep 06 2023

Formula

G.f.: (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)/( 528*t^16 - 32*t^15 - 32*t^14 - 32*t^13 - 32*t^12 - 32*t^11 - 32*t^10 - 32*t^9 - 32*t^8 - 32*t^7 - 32*t^6 - 32*t^5 - 32*t^4 - 32*t^3 - 32*t^2 - 32*t + 1).

From G. C. Greubel, Sep 06 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 33*t + 560*t^16 - 528*t^17).

a(n) = 32*Sum_{j=1..15} a(n-j) - 528*a(n-16). (End)