A164375 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359260, 18873792, 150988068, 1207886400, 9662946048, 77302407168, 618409967616, 4947205424364, 39577048871472, 316611634855572, 2532855030486480, 20262535861599360, 162097851871033344

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..1000

Index entries for linear recurrences with constant coefficients, signature (7,7,7,7,7,7,-28).

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(28*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

a(n) = -28*a(n-7) + 7*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 17 2017 *)
coxG[{7, 28, -7}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 20 2019 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8)) \\ G. C. Greubel, Sep 17 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8) )); // G. C. Greubel, Aug 10 2019
(Sage)
def A164375_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8)).list()
A164375_list(30) # G. C. Greubel, Aug 10 2019
(GAP) a:=[9, 72, 576, 4608, 36864, 294912, 2359260];; for n in [8..30] do a[n]:=7*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -28*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Crossrefs

Sequence in context: A162960 A163391 A163953 * A164777 A165216 A165787

Adjacent sequences: A164372 A164373 A164374 * A164376 A164377 A164378

Keyword

nonn

Author

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

Status

approved