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A081068
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(Lucas(4n+2)+2)/5, or Fibonacci(2n+1)^2, or A081067/5.
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4
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1, 4, 25, 169, 1156, 7921, 54289, 372100, 2550409, 17480761, 119814916, 821223649, 5628750625, 38580030724, 264431464441, 1812440220361, 12422650078084, 85146110326225, 583600122205489, 4000054745112196, 27416783093579881
(list;
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refs;
listen;
history;
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internal format)
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OFFSET
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0,2
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REFERENCES
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Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 19.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (8,-8,1).
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FORMULA
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a(n) = 8a(n-1)-8a(n-2)+a(n-3).
a(n) = (2/5)+(3/10)*{[(7/2)-(3/2)*sqrt(5)]^n+[(7/2)+(3/2)*sqrt(5)]^n}+(1/10)*sqrt(5)*{[(7/2) +(3/2)*sqrt(5)]^n-[(7/2)-(3/2)*sqrt(5)]^n}, with n>=0. [Paolo P. Lava, Dec 01 2008]
a(n) = Fibonacci(2*n)*Fibonacci(2*n+2)+1. [Gary Detlefs, Apr 01 2012]
G.f.: -(1-4*x+x^2)/((x-1)*(x^2-7*x+1)). [Colin Barker, Jun 26 2012]
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MAPLE
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luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d, `, (luc(4*n+2)+2)/5) od:
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MATHEMATICA
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CoefficientList[Series[-(1-4*x+x^2)/((x-1)*(x^2-7*x+1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{8, -8, 1}, {1, 4, 25}, 50] (* Vincenzo Librandi, Jun 26 2012 *)
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PROG
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(MAGMA) I:=[1, 4, 25]; [n le 3 select I[n] else 8*Self(n-1)-8*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012
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CROSSREFS
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Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081067.
Equals A001519(n)^2 and A058038 - 1.
First differences of A103433.
Sequence in context: A225137 A006880 A175255 * A163072 A140177 A034494
Adjacent sequences: A081065 A081066 A081067 * A081069 A081070 A081071
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KEYWORD
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nonn,easy
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AUTHOR
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R. K. Guy, Mar 04, 2003
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EXTENSIONS
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More terms and Maple code from James A. Sellers, Mar 05, 2003
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STATUS
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approved
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