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A164697
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Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.
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1
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1, 4, 12, 36, 108, 324, 972, 2916, 8742, 26208, 78576, 235584, 706320, 2117664, 6349104, 19035648, 57071982, 171111132, 513019140, 1538115228, 4611520836, 13826093148, 41452886916, 124282529820, 372619336494, 1117173669768
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OFFSET
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0,2
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COMMENTS
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The initial terms coincide with those of A003946, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
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LINKS
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FORMULA
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G.f.: (x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/( 3*x^8 - 2*x^7 - 2*x^6 - 2*x^5 - 2*x^4 - 2*x^3 - 2*x^2 - 2*x + 1).
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MAPLE
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seq(coeff(series((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 16 2019
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MATHEMATICA
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CoefficientList[Series[(x^8 +2x^7 +2x^6 +2x^5 +2x^4 +2x^3 +2x^2 +2x +1)/( 3x^8 -2x^7 -2x^6 -2x^5 -2x^4 -2x^3 -2x^2 -2x +1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
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PROG
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(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9)) \\ G. C. Greubel, Sep 16 2019
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9) )); // G. C. Greubel, Sep 16 2019
(Sage)
P.<t> = PowerSeriesRing(ZZ, prec)
return P((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9)).list()
(GAP) a:=[4, 12, 36, 108, 324, 972, 2916, 8742];; for n in [9..30] do a[n]:=2*Sum([1..7], j-> a[n-j]) -3*a[n-8]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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