A166437 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 43, 1806, 75852, 3185784, 133802928, 5619722976, 236028364992, 9913191329664, 416354035845888, 17486869505527296, 734448519232145529, 30846837807750074292, 1295567187925501528275, 54413821892870997324012

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170762, 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 (41,41,41,41,41,41,41,41,41,41,-861).

Crossrefs

Cf. A154638, A169452, A170762.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
f:= func< p,q,x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;
Coefficients(R!( f(861,41,x) )); // G. C. Greubel, Jul 26 2024

Mathematica

With[{p=861, q=41}, 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 14 2016; Jul 26 2024 *)
coxG[{11, 861, -41, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 26 2024 *)

SageMath

def f(p,q,x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
def A166437_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(861,41,x) ).list()
A166437_list(30) # G. C. Greubel, Jul 26 2024

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)/(861*t^11 - 41*t^10 - 41*t^9 - 41*t^8 - 41*t^7 - 41*t^6 - 41*t^5 - 41*t^4 - 41*t^3 - 41*t^2 - 41*t + 1).

a(n) = -861*a(n-11) + 41*Sum_{k=1..10} a(n-k). - Wesley Ivan Hurt, Mar 17 2023

G.f.: (1+x)*(1-x^11)/(1 - 42*x + 902*x^11 - 861*x^12). - G. C. Greubel, Jul 26 2024