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A162740
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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)^3 = I.
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2
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1, 4, 12, 30, 72, 168, 390, 900, 2076, 4782, 11016, 25368, 58422, 134532, 309804, 713406, 1642824, 3783048, 8711526, 20060676, 46195260, 106377294, 244963080, 564094968, 1298984214, 2991269124, 6888221772, 15862029150, 36526694472, 84112781928, 193692865350
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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.
Also, expansion of b(2)*b(3)/(1 - 2*x - 2*x^2 + 3*x^3), where b(k) = (1-x^k)/(1-x).
This is also the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_22 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).
(End)
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LINKS
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FORMULA
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G.f.: (x^3 + 2*x^2 + 2*x + 1)/(3*x^3 - 2*x^2 - 2*x + 1).
a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>3.
a(n) = -2 + ((-7+2*sqrt(13))*(1-sqrt(13))^n + (7+2*sqrt(13))*(1+sqrt(13))^n)/(3*sqrt(13)*2^(n-1)) for n>0. (End)
G.f.: (1+x)*(1-x^3)/(1 -3*x +5*x^3 -3*x^4). - G. C. Greubel, Apr 25 2019
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MATHEMATICA
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CoefficientList[Series[(x^3+2x^2+2x+1)/(3x^3-2x^2-2x+1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
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PROG
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(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)*b(3)/(1-2*x-2*x^2+3*x^3))); // Bruno Berselli, Dec 28 2015 - see Chapovalov et al.
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4) )); // G. C. Greubel, Apr 25 2019
(PARI) my(x='x+O('x^40)); Vec((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)) \\ G. C. Greubel, Apr 25 2019
(Sage) ((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019
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CROSSREFS
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Cf. similar sequences listed in A265055.
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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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