A168680 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196605, 393204, 786399, 1572780, 3145524, 6290976, 12581808, 25163328, 50326080, 100651008, 201299712, 402594816, 805180416, 1610342400, 3220647936

Offset

0,2

Comments

The initial terms coincide with those of A003945, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 196605, A003945(17) = 196608. - Klaus Brockhaus, Mar 30 2011

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 (2,-1,2,-1,2,-1,2,-1,2,-1,2,-1,2,-1,2,-1).

Formula

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

G.f.: (1+t)*(1-t^17)/(1 -2*t +2*t^17 -t^18). - G. C. Greubel, Feb 22 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -2*t +2*t^17 -t^18), {t, 0, 40}], t] (* G. C. Greubel, Jul 29 2016, Feb 22 2021 *)

Prog

(PARI) Vec(Pol(vector(17, i, 1))/Pol(vector(17, i, if(i%2, 1, -2)))+O(x^99)) \\ Charles R Greathouse IV, Jul 30 2016
(Magma)
R<t>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -2*t +2*t^17 -t^18) )); // G. C. Greubel, Feb 22 2021
(SageMath)
def A168680_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -2*t +2*t^17 -t^18) ).list()
A168680_list(40) # G. C. Greubel, Feb 22 2021

Crossrefs

Cf. A003945 (G.f.: (1+x)/(1-2*x)).

Sequence in context: A167104 A167648 A167881 * A168728 A168776 A168824

Adjacent sequences: A168677 A168678 A168679 * A168681 A168682 A168683

Keyword

easy,nonn

Author

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

Status

approved