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A164373
Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
1
1, 8, 56, 392, 2744, 19208, 134456, 941164, 6587952, 46114320, 322790832, 2259469968, 15815828784, 110707574544, 774930433956, 5424354927432, 37969377752376, 265777897314888, 1860391054122552, 13022357800350024
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003950, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
FORMULA
G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).
a(n) = -21*a(n-7) + 6*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021
MAPLE
seq(coeff(series((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 17 2017 *)
coxG[{7, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)
PROG
(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)) \\ G. C. Greubel, Sep 17 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8) )); // G. C. Greubel, Aug 28 2019
(Sage)
def A164373_list(prec):
P.<t> = PowerSeriesRing(ZZ, prec)
return P((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)).list()
A164373_list(30) # G. C. Greubel, Aug 28 2019
(GAP) a:=[8, 56, 392, 2744, 19208, 134456, 941164];; for n in [8..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -21*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019
CROSSREFS
Sequence in context: A234274 A163347 A163924 * A164769 A165215 A165786
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved