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A166467
Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1
1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6141, 12276, 24543, 49068, 98100, 196128, 392112, 783936, 1567296, 3133440, 6264576, 12524544, 25039878, 50061345, 100085880, 200098167, 400049202, 799804248, 1599020400, 3196865040
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003945, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
FORMULA
G.f.: (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)/(t^12 - t^11 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2 - t + 1).
From G. C. Greubel, Jul 28 2024: (Start)
a(n) = Sum_{j=1..11} a(n-j) - a(n-11).
G.f.: (1+x)*(1-x^12)/(1 - 2*x + 2*x^12 - x^13). (End)
MATHEMATICA
With[{p=1, q=1}, CoefficientList[Series[(1+t)*(1-t^12)/(1-(q+1)*t + (p+ q)*t^12-p*t^13), {t, 0, 40}], t]] (* G. C. Greubel, May 15 2016; Jul 28 2024 *)
coxG[{12, 1, -1, 40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 15 2020 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
f:= func< p, q, x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
Coefficients(R!( f(1, 1, x) )); // G. C. Greubel, Jul 28 2024
(SageMath)
def f(p, q, x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166467_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(1, 1, x) ).list()
A166467_list(30) # G. C. Greubel, Jul 28 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved