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

0,2

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).

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)

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 *)

Prog

(Magma)
R<x>:=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
(SageMath)
def A166599_list(prec):
    P.<x> = 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

Crossrefs

Cf. A154638, A169452, A170737.

Sequence in context: A165329 A165880 A166412 * A167048 A167125 A167674

Adjacent sequences: A166596 A166597 A166598 * A166600 A166601 A166602

Keyword

nonn

Author

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

Status

approved