A167896 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709110, 21474836400, 85899345450, 343597381200, 1374389522400, 5497558080000, 21990232281600, 87960928972800

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003947, 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 (3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,-6).

Crossrefs

Cf. A003947, A154639, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-4*x+9*x^16-6*x^17) )); // G. C. Greubel, Dec 06 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-4*t+9*t^16-6*t^17), {t,0,50}], t] (* G. C. Greubel, Jul 01 2016; Dec 06 2024 *)
coxG[{16,6,-3,40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 06 2024 *)

SageMath

def A167896_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-4*x+9*x^16-6*x^17) ).list()
print(A167896_list(40)) # G. C. Greubel, Dec 06 2024

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

From G. C. Greubel, Dec 06 2024: (Start)

a(n) = 3*Sum_{j=1..15} a(n-j) - 6*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 4*x + 9*x^16 - 6*x^17). (End)