A167048 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

1, 18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776946, 36287890208082, 616894133537394, 10487200270135545, 178282404592301664, 3030800878069084224, 51523614927173682720

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, 16, -136).

Formula

G.f.: (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)/(136*t^13 - 16*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).

G.f.: (1+x)*(1-x^13)/(1 - 17*x + 152*x^13 - 136*x^14). - G. C. Greubel, Apr 26 2019

a(n) = -136*a(n-13) + 16*Sum_{k=1..12} a(n-k). - Wesley Ivan Hurt, May 06 2021

Mathematica

CoefficientList[Series[(1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14), {x, 0, 20}], x] (* G. C. Greubel, May 30 2016, modified Apr 26 2019 *)
coxG[{13, 136, -16}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14)) \\ G. C. Greubel, Apr 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14) )); // G. C. Greubel, Apr 26 2019
(Sage) ((1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

Crossrefs

Sequence in context: A165880 A166412 A166599 * A167125 A167674 A167927

Adjacent sequences: A167045 A167046 A167047 * A167049 A167050 A167051

Keyword

nonn

Author

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

Status

approved