A069202 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.

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

Offset: 1

Author

Benoit Cloitre, Apr 11 2002

Keywords

• easy
• nonn

Comments

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.

Examples

			a(1)=1 and a(2)=2 have different parities, hence a(3)=a(2)+a(1)=3.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..1000
• A. M. Amleh et al., On Some Difference Equations with Eventually Periodic Solutions, J. Math. Anal. Appl., 223 (1998), 196-215. [_N. J. A. Sloane_, Jun 17 2009]
• J. Greene, The Unboundedness of a Family of Difference Equations Over the Integers, Fib. Q., 46/47 (2008/2009), 146-152. [_N. J. A. Sloane_, Jun 17 2009]

Crossrefs

Cf. A069162, A151749.

Programs

Mathematica

Nest[Append[#, If[OddQ[#], #, #/2] &@(#[[-1]] + #[[-2]])] &, {1, 2}, 47] (* Ivan Neretin, Sep 07 2017 *)

Formula

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.