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

Author

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

Keywords

• nonn

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).

Crossrefs

Cf. A003953, A154638, A169452.

Programs

Magma

R:=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

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 *)

SageMath

def A167914_list(prec):
    P. = 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

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)