A167916 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520652, 414272545727172, 4556998002998892, 50126978032987746, 551396758362864480

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003954, 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 (10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,-55).

Crossrefs

Cf. A003954, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^16)/(1-11*x+65*x^16-55*x^17) )); // G. C. Greubel, Nov 10 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-11*t+65*t^16-55*t^17), {t,0,50}], t] (* G. C. Greubel, Jul 01 2016; Nov 10 2023 *)
coxG[{16,55,-10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Nov 10 2023 *)

SageMath

def A167916_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-11*x+65*x^16-55*x^17) ).list()
A167916_list(30) # G. C. Greubel, Nov 10 2023

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)/( 55*t^16 - 10*t^15 - 10*t^14 - 10*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

From G. C. Greubel, Nov 10 2023: (Start)

a(n) = 10*Sum_{j=1..15} a(n-j) - 55*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 11*x + 65*x^16 - 55*x^17). (End)