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A106568
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Expansion of 4*x/(1 - 4*x - 4*x^2).
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1
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0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360
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OFFSET
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0,2
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COMMENTS
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This sequence is part of a class of sequences with the properties: a(n) = m*(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = m, g.f.: m*x/(1 - m*x - m*x^2), and have the Binet form m*(alpha^n - beta^n)/(alpha - beta) where 2*alpha = m + sqrt(m^2 + 4*m) and 2*beta = p - sqrt(m^2 + 4*m). - G. C. Greubel, Sep 06 2021
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LINKS
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FORMULA
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a(n) = 4*a(n-1) + 4*a(n-2) for n > 2; a(0) = 0, a(1)=4.
G.f.: 4*x/(1 - 4*x - 4*x^2). (End)
G.f.: Q(0) - 1, where Q(k) = 1 + 2*(1+2*x)*x + 2*(2*k+3)*x - 2*x*(2*k+1 +2*x+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
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MATHEMATICA
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LinearRecurrence; {4, 4}, {0, 4}, 40] (* G. C. Greubel, Sep 06 2021 *)
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PROG
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(Magma) [n le 2 select 4*(n-1) else 4*(Self(n-1) +Self(n-2)): n in [1..41]]; // G. C. Greubel, Sep 06 2021
(Sage) [2^(n+1)*lucas_number1(n, 2, -1) for n in (0..40)] # G. C. Greubel, Sep 06 2021
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CROSSREFS
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KEYWORD
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nonn,easy,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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