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].