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A163063
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Lucas(3n+2) = Fib(3n+1) + Fib(3n+3).
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3
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3, 11, 47, 199, 843, 3571, 15127, 64079, 271443, 1149851, 4870847, 20633239, 87403803, 370248451, 1568397607, 6643838879, 28143753123, 119218851371, 505019158607, 2139295485799, 9062201101803, 38388099893011
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.
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LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 0..400
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FORMULA
| a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.
G.f.: (3-x)/(1-4*x-x^2).
a(n) = A033887(n) + A014445(n+1).
a(n) = ((3+sqrt(5))*(2+sqrt(5))^n+(3-sqrt(5))*(2-sqrt(5))^n)/2.
a(n)= A000032(3*n+2), n>=0, (Lucas trisection). W Lang, Mar 09 2011.
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MAPLE
| with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # Nathaniel Johnston, Apr 18 2011
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MATHEMATICA
| Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* From Alonso del Arte (alonso.delarte(AT)gmail.com), Nov 29 2010 *)
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PROG
| (MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 21 2009]
(MAGMA) [Lucas(3*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
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CROSSREFS
| Cf. A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).
Sequence in context: A030814 A030976 A112567 * A151142 A151143 A151144
Adjacent sequences: A163060 A163061 A163062 * A163064 A163065 A163066
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KEYWORD
| nonn
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AUTHOR
| Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009
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EXTENSIONS
| Edited and extended beyond a(5) by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 21 2009
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