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A153594
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a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).
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6
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1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808
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OFFSET
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1,2
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COMMENTS
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Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.
Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....
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LINKS
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FORMULA
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G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
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MATHEMATICA
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LinearRecurrence[{8, -13}, {1, 8}, 40] (* Harvey P. Dale, Aug 16 2012 *)
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PROG
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(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
(Sage) [lucas_number1(n, 8, 13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009
(Magma) I:=[1, 8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016
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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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Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
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EXTENSIONS
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STATUS
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approved
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