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A097512
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a(n) = 6*Lucas(2n) - Fibonacci(2n+2).
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1
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11, 15, 34, 87, 227, 594, 1555, 4071, 10658, 27903, 73051, 191250, 500699, 1310847, 3431842, 8984679, 23522195, 61581906, 161223523, 422088663, 1105042466, 2893038735, 7574073739, 19829182482, 51913473707, 135911238639
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OFFSET
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0,1
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COMMENTS
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Sequence relates bisections of Lucas and Fibonacci numbers.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).
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FORMULA
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a(n) = 8*Lucas(2n) - Lucas(2n+2) - 2*Fibonacci(2n-1) = 8*A005248(n) - A005248(n+1) - 2*A001519(n).
a(n+1)/a(n) approaches the golden ratio phi + 1 = (3+sqrt(5))/2.
From Philippe Deléham, Nov 16 2008: (Start)
a(n) = 3*a(n-1) - a(n-2) with a(0)=11 and a(1)=15.
G.f.: (11-18*x)/(1-3*x+x^2). (End)
a(n) = (11/2)*( ((3+sqrt(5))/2)^n + ((3-sqrt(5))/2)^n ) - (3*sqrt(5)/10)*( ((3+sqrt(5))/2)^n + ((3-sqrt(5)/2)^n ), with n>=0. - Paolo P. Lava, Nov 19 2008
a(n) = 4*Fibonacci(2n+1) + 7*Fibonacci(2n-1) = 4*Lucas(2n) + 3*Fibonacci(2n-1). - Ron Knott, Jul 01 2013
E.g.f.: exp(3*x/2)*( 11*cosh(sqrt(5)*x/2) - (3/sqrt(5))*sinh(sqrt(5)*x/2) ). - G. C. Greubel, Jan 19 2020
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MAPLE
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f:=fibonacci; f[combinat]; seq(4*f(2*n+1)+7*f(2*n-1), n=0..30); # G. C. Greubel, Jan 19 2020
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MATHEMATICA
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Table[6LucasL[2n]-Fibonacci[2n+2], {n, 0, 30}] (* or *) LinearRecurrence[ {3, -1}, {11, 15}, 30] (* Harvey P. Dale, Oct 02 2011 *)
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PROG
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(Magma) [8*Lucas(2*n) - Lucas(2*n+2) - 2*Fibonacci(2*n-1): n in [0..30]]; // Vincenzo Librandi, Oct 03 2011
(PARI) vector(31, n, 4*fibonacci(2*n-1) +7*fibonacci(2*n-3) ) \\ G. C. Greubel, Jan 19 2020
(Sage) [6*lucas_number2(2*n, 1, -1) - fibonacci(2*n+2) for n in (0..30)] # G. C. Greubel, Jan 19 2020
(GAP) List([0..30], n-> 6*Lucas(1, -1, 2*n)[2] - Fibonacci(2*n+2) ); # G. C. Greubel, Jan 19 2020
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CROSSREFS
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Cf. A000032, A000045, A005248, A022133.
Sequence in context: A030099 A085597 A316643 * A032490 A068483 A357626
Adjacent sequences: A097509 A097510 A097511 * A097513 A097514 A097515
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KEYWORD
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nonn,easy
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AUTHOR
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Creighton Dement, Aug 26 2004
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EXTENSIONS
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New definition from Ralf Stephan, Dec 01 2004
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STATUS
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approved
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