A167914 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1100000000000, 11000000000000, 110000000000000, 1100000000000000, 10999999999999945, 109999999999998900

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 (9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,-45).

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)/( 45*t^16 - 9*t^15 - 9*t^14 - 9*t^13 - 9*t^12 - 9*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, Dec 04 2024: (Start)

a(n) = 9*Sum_{j=1..15} a(n-j) - 45*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 10*x + 54*x^16 - 45*x^17). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-10*t+54*t^16-45*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Dec 04 2024 *)
coxG[{16, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 04 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^16)/(1-10*x+54*x^16-45*x^17) )); // G. C. Greubel, Dec 04 2024
(SageMath)
def A167914_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-10*x+54*x^16-45*x^17) ).list()
A167914_list(40) # G. C. Greubel, Dec 04 2024

Crossrefs

Cf. A003953, A154638, A169452.

Sequence in context: A166950 A167112 A167664 * A003953 A168688 A168736

Adjacent sequences: A167911 A167912 A167913 * A167915 A167916 A167917

Keyword

nonn

Author

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

Status

approved