A167949 Number of reduced words of length n in Coxeter group on 33 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 33, 1056, 33792, 1081344, 34603008, 1107296256, 35433480192, 1133871366144, 36283883716608, 1161084278931456, 37154696925806592, 1188950301625810944, 38046409652025950208, 1217485108864830406656, 38959523483674573012992

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170752, 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 (31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,-496).

Crossrefs

Cf. A154638, A169452, A170752.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-32*x+527*x^16-496*x^17) )); // G. C. Greubel, Sep 07 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-32*t+527*t^16-496*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 07 2023 *)
coxG[{16,496,-31}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 22 2020 *)

SageMath

def A167949_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-32*x+527*x^16-496*x^17) ).list()
A167949_list(40) # G. C. Greubel, Sep 07 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)/( 496*t^16 - 31*t^15 - 31*t^14 - 31*t^13 - 31*t^12 - 31*t^11 - 31*t^10 - 31*t^9 - 31*t^8 - 31*t^7 - 31*t^6 - 31*t^5 - 31*t^4 - 31*t^3 - 31*t^2 - 31*t + 1).

From G. C. Greubel, Sep 07 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 32*t + 527*t^16 - 496*t^17).

a(n) = 31*Sum_{j=1..15} a(n-j) - 496*a(n-16). (End)