cp's OEIS Frontend

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.

%I A069202 #20 Jan 05 2025 19:51:36
%S A069202 1,2,3,5,4,9,13,11,12,23,35,29,32,61,93,77,85,81,83,82,165,247,206,
%T A069202 453,659,556,1215,1771,1493,1632,3125,4757,3941,4349,4145,4247,4196,
%U A069202 8443,12639,10541,11590,22131,33721,27926,61647,89573,75610,165183,240793
%N 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.
%C A069202 A Collatz-Fibonacci mixture. Does this sequence diverge to infinity? [Yes! See Amleh et al. - _N. J. A. Sloane_, Jun 17 2009]
%C A069202 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.
%H A069202 Ivan Neretin, <a href="/A069202/b069202.txt">Table of n, a(n) for n = 1..1000</a>
%H A069202 A. M. Amleh et al., <a href="https://doi.org/10.1006/jmaa.1998.5971">On Some Difference Equations with Eventually Periodic Solutions</a>, J. Math. Anal. Appl., 223 (1998), 196-215. [_N. J. A. Sloane_, Jun 17 2009]
%H A069202 J. Greene, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-2/Greene_1-09.pdf">The Unboundedness of a Family of Difference Equations Over the Integers</a>, Fib. Q., 46/47 (2008/2009), 146-152. [_N. J. A. Sloane_, Jun 17 2009]
%F A069202 a(n+2) = 2*(a(n+1) + a(n))/(3 + (-1)^(a(n+1) + a(n))).
%F A069202 It seems that a(n)*exp(-n/4) is bounded.
%e A069202 a(1)=1 and a(2)=2 have different parities, hence a(3)=a(2)+a(1)=3.
%t A069202 Nest[Append[#, If[OddQ[#], #, #/2] &@(#[[-1]] + #[[-2]])] &, {1, 2}, 47] (* _Ivan Neretin_, Sep 07 2017 *)
%Y A069202 Cf. A069162, A151749.
%K A069202 easy,nonn
%O A069202 1,2
%A A069202 _Benoit Cloitre_, Apr 11 2002