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A022086
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Fibonacci sequence beginning 0, 3.
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27
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0, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507
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OFFSET
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0,2
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COMMENTS
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Pisano period lengths: 1, 3, 1, 6, 20, 3, 16, 12, 8, 60, 10, 6, 28, 48, 20, 24, 36, 24, 18, 60, ... . - R. J. Mathar, Aug 10 2012
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 7,17.
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LINKS
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FORMULA
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a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - Thomas Baruchel, Sep 08 2004
a(n) = 3*F(n). Also, a(n) = F(n-2) + F(n+2) for n>1, with F=A000045.
E.g.f.: 6*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - Ilya Gutkovskiy, Oct 29 2016
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MAPLE
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BB := n->if n=0 then 3; > elif n=1 then 0; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 1 to 34 do L:=[op(L), BB(k)]: od: L; # Zerinvary Lajos, Mar 19 2007
with (combinat):seq(sum((fibonacci(n, 1)), m=1..3), n=0..32); # Zerinvary Lajos, Jun 19 2008
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MATHEMATICA
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Table[Fibonacci[n + 4] + Fibonacci[n - 4] - 4 Fibonacci[n], {n, 0, 40}] (* Bruno Berselli, Dec 30 2016 *)
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PROG
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CROSSREFS
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Cf. sequences with formula Fibonacci(n+k)+Fibonacci(n-k) listed in A280154.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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