A166369 - OEIS

A166369

Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 109999999945, 1099999998900, 10999999983555, 109999999781100, 1099999997266500, 10999999967220000, 109999999617750000

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A003953, 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 (9,9,9,9,9,9,9,9,9,9,-45).

FORMULA

G.f.: (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)/(45*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

From G. C. Greubel, Jul 23 2024: (Start)

a(n) = 9*Sum_{j=1..10} a(n-j) - 45*a(n-11).

G.f.: (1+t)*(1-t^11)/(1 - 10*t + 54*t^11 - 45*t^12). (End)

MATHEMATICA

With[{p=45, q=9}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t, 0, 40}], t]] (* G. C. Greubel, May 10 2016; Jul 23 2024 *)

coxG[{11, 45, -9}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 12 2016 *)

PROG

(Magma)

R<x>:=PowerSeriesRing(Integers(), 30);

Coefficients(R!( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) )); // G. C. Greubel, Jul 23 2024

(SageMath)

def A166369_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) ).list()

A166369_list(30) # G. C. Greubel, Jul 23 2024

CROSSREFS

Cf. A003953, A154638, A169452.