A167881 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98301, 196596, 393183, 786348, 1572660, 3145248, 6290352, 12580416, 25160256, 50319360, 100636416, 201268224, 402527232, 805036032, 1610035200, 3219996672

Offset: 0

Author

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

Keywords

• nonn

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..500
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1).

Crossrefs

Cf. A003945, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-2*x+2*x^16-x^17) )); // G. C. Greubel, Dec 06 2024

Mathematica

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

SageMath

def A167881_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-2*x+2*x^16-x^17) ).list()
print(A167881_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)/(t^16 - t^15 - t^14 - t^13 - t^12 - t^11 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2 - t + 1).

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

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

G.f.: (1+x)*(1-x^16)/(1 - 2*x + 2*x^16 - x^17). (End)