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A069202
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A Collatz-Fibonacci mixture: a(1) = 1, a(2) = 2, a(n+2) = a(n+1)/2+a(n)/2 if a(n+1) and a(n) have the same parity, a(n+2) = a(n+1)+a(n) otherwise.
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5
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1, 2, 3, 5, 4, 9, 13, 11, 12, 23, 35, 29, 32, 61, 93, 77, 85, 81, 83, 82, 165, 247, 206, 453, 659, 556, 1215, 1771, 1493, 1632, 3125, 4757, 3941, 4349, 4145, 4247, 4196, 8443, 12639, 10541, 11590, 22131, 33721, 27926, 61647, 89573, 75610, 165183, 240793
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OFFSET
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1,2
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COMMENTS
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A Collatz-Fibonacci mixture. Does this sequence diverge to infinity? [Yes! See Amleh et al. - N. J. A. Sloane, Jun 17 2009]
Conjecture: More generally, let a(1)=x and a(2)=y be two distinct positive integers; then for any x,y > 0, lim n -> infinity log(a(n))/n = 1/4.
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LINKS
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FORMULA
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a(n+2) = 2*(a(n+1) + a(n))/(3 + (-1)^(a(n+1) + a(n))).
It seems that a(n)*exp(-n/4) is bounded.
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EXAMPLE
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a(1)=1 and a(2)=2 have different parities, hence a(3)=a(2)+a(1)=3.
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MATHEMATICA
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Nest[Append[#, If[OddQ[#], #, #/2] &@(#[[-1]] + #[[-2]])] &, {1, 2}, 47] (* Ivan Neretin, Sep 07 2017 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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