A167900 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959552, 9663676416, 77309411328, 618475290624, 4947802324992, 39582418599936, 316659348799452, 2533274790395328, 20266198323160356, 162129586585264704

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003951, 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 (7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,-28).

Crossrefs

Cf. A003951, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-8*x+35*x^16-28*x^17) )); // G. C. Greubel, Dec 06 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-8*t+35*t^16-28*t^17), {t,0,50}], t] (* G. C. Greubel, Jul 01 2016; Dec 06 2024 *)
coxG[{16,28,-7}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 06 2024 *)

SageMath

def A167900_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-8*x+35*x^16-28*x^17) ).list()
print(A167900_list(40)) # G. C. Greubel, Dec 06 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)/( 28*t^16 - 7*t^15 - 7*t^14 - 7*t^13 - 7*t^12 - 7*t^11 - 7*t^10 - 7*t^9 - 7*t^8 - 7*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

From G. C. Greubel, Dec 06 2024: (Start)

a(n) = 7*Sum_{j=1..15} a(n-j) - 28*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 8*x + 35*x^16 - 28*x^17). (End)