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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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3
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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
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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.
First differences are in A161728.
lim_{n -> infinity} a(n)/a(n-1) = 4+sqrt(3) = 5.73205080756887729....
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LINKS
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Table of n, a(n) for n=1..21.
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FORMULA
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G.f.: x/(1-8*x+13*x^2). [From 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. [From Philippe DELEHAM, Jan 01 2009]
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MATHEMATICA
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Join[{a=1, b=8}, Table[c=8*b-13*a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 19 2011*)
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] ]; [From Klaus Brockhaus, Dec 31 2008]
(Sage) [lucas_number1(n, 8, 13) for n in xrange(1, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
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Cf. A002194 (decimal expansion of sqrt(3)), A054491, A162766, A074324, A161728.
Sequence in context: A034516 A069325 A082135 * A037697 A037606 A055147
Adjacent sequences: A153591 A153592 A153593 * A153595 A153596 A153597
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KEYWORD
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nonn
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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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Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008
Edited by Klaus Brockhaus, Oct 11 2009
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STATUS
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approved
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