A166541 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959552, 9663676416, 77309411292, 618475290048, 4947802318116, 39582418526784, 316659348069120, 2533274783391744, 20266198257844224, 162129585988435968

Offset

0,2

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,-28).

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)/(28*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, Aug 23 2024: (Start)

a(n) = 7*Sum_{j=1..11} a(n-j) - 28*a(n-13).

G.f.: (1+x)*(1-x^12)/(1 - 8*x + 35*x^12 - 28*x^13). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-8*t+35*t^12-28*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 16 2016; Aug 23 2024 *)
coxG[{12, 28, -7, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 23 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-8*x+35*x^12-28*x^13) )); // G. C. Greubel, Aug 23 2024
(SageMath)
def A166541_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-8*x+35*x^12-28*x^13) ).list()
A166541_list(30) # G. C. Greubel, Aug 23 2024
(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; -28, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7]^(n-1)*[9; 72; 576; 4608; 36864; 294912; 2359296; 18874368; 150994944; 1207959552; 9663676416; 77309411292])[1, 1], 1) \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A003951, A154638, A169452.

Sequence in context: A165216 A165787 A166367 * A166924 A167110 A167658

Adjacent sequences: A166538 A166539 A166540 * A166542 A166543 A166544

Keyword

nonn,easy,changed

Author

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

Status

approved