Half-Fib Sequence

I discovered an integer sequence that I call the Half-Fib sequence.  I call it the Half-Fib sequence, because it is like the Fibonacci sequence, except that terms are halved before being added, if they are even. In order to make this sequence work, the sequence cannot start with 1,1 or 1,2. Instead, 1,3, can be chosen to produce:

1, 3, 4, 5, 7, 12, 13, 19, 32, 35, 51, 86, 94, 90, 92, 91, 137, 228, 251, 365, 616, 673, 981, 1654, 1808, 1731, 2635, 4366, 4818, 4592, 4705, 7001, 11706, 12854, 12280, 12567, 18707, 31274, 34344, 32809, 49981, 82790, 91376, 87083, 132771, 219854, 242698, 231276, 236987, 352625, 589612, 647431, 942237, 1589668, 1737071, 2531905, 4268976, 4666393, 6800881, 11467274, 12534518, 12000896, 12267707, 18268155, 30535862, 33536086, 32035974, 32786030, 32411002, 32598516, 32504759, 48804017, 81308776, 89458405, 130112793, 219571198, 239898392, 229734795, 349683991, 579418786, 639393384, 609406085, 929102777, 1538508862, 1698357208, 1618433035, 2467611639, 4086044674, 4510633976, 4298339325, 6553656313, 10851995638, 11979654132, 11415824885, 17405651951, 28821476836, 31816390369, 46227128787, 78043519156

For example,

1 + 3 = 4

3 + 4/2 = 5

4/2 + 5 = 7

5 + 7 = 12

7 + 12/2 = 13

This sequence is similar to the 3x+1 sequence in that the pattern of even and odd numbers is somewhat unpredictable. It also seems similar to the classic Fibonacci sequence, in that the terms appear to grow in a roughly geometric way. 

If you want to generate the sequence in Mathematica, you can use the Module:

HalfFib[a_, b_, n_] := Module[{HF, i},
HF = {a, b};
For [i = 3, i < n, i++,
HF = 
Append[HF, 
HF[[i - 2]]/(2 - Mod[HF[[i - 2]], 2]) + 
HF[[i - 1]]/(2 - Mod[HF[[i - 1]], 2])]];
HF]

Using this Mathematica module, I was able to generate the above sequence with HalfFib[1,3,100].

Fact

  1. For Half-Fib sequences that start with a pair of unique positive integers other than {1,2}, or {2,1} the sequence has no maximum value.

Conjectures

  1. If the sequence grows indefinitely, then the ratio of even terms to odd terms is 1:1 in the limit.
  2. Like a random walk, there is no maximum difference between the number of even and odd terms in the initial subsequences.