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

0,2

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).

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)

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 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-2*x+2*x^16-x^17) )); // G. C. Greubel, Dec 06 2024
(SageMath)
def A167881_list(prec):
    P.<x> = 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

Crossrefs

Cf. A003945, A154638, A169452.

Sequence in context: A364496 A167104 A167648 * A168680 A168728 A168776

Adjacent sequences: A167878 A167879 A167880 * A167882 A167883 A167884

Keyword

nonn

Author

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

Status

approved