A167882 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395622, 172186848, 516560496, 1549681344, 4649043600, 13947129504, 41841384624, 125524142208, 376572391632, 1129717069920

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003946, 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 (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,-3).

Crossrefs

Cf. A003946, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-3*x+5*x^16-3*x^17) )); // G. C. Greubel, Dec 06 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-3*t+5*t^16-3*t^17), {t,0,50}], t] (* G. C. Greubel, Jun 29 2016; Dec 06 2024 *)
coxG[{16,3,-2}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 06 2024 *)

SageMath

def A167882_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-3*x+5*x^16-3*x^17) ).list()
print(A167882_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) / ( 3*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).

From G. C. Greubel, Jan 17 2023: (Start)

a(n) = 2*Sum_{j=1..15} a(n-j) - 3*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 3*x + 5*x^16 - 3*x^17). (End)