A166599 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776946, 36287890208082, 616894133537241, 10487200270130496, 178282404592174368, 3030800878066215168, 51523614927112923360

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170737, 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 (16,16,16,16,16,16,16,16,16,16,16,-136).

Crossrefs

Cf. A154638, A169452, A170737.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^12)/(1 - 17*x+152*x^12-136*x^13) )); // G. C. Greubel, Dec 08 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-17*t+152*t^12-136*t^13), {t,0,50}], t] (* G. C. Greubel, May 18 2016; Dec 08 2024 *)
coxG[{12,136,-16,40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 08 2024 *)

SageMath

def A166599_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-17*x+152*x^12-136*x^13) ).list()
print(A166599_list(40)) # G. C. Greubel, Dec 08 2024

Formula

G.f.: (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)/(136*t^12 - 16*t^11 - 16*t^10 - 16*t^9 -16*t^8 - 16*t^7 - 16*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 -16*t +1).

From G. C. Greubel, Dec 08 2024: (Start)

a(n) = 16*Sum_{j=1..11} a(n-j) - 136*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 17*x + 152*x^12 - 136*x^13). (End)