By the magic of taking a dump with a notepad, I solved the first sequence of stage 4:
2, 4, 8, 1, 3, 6, 18, 26, ?, 12, 24, 49, 89, 134, 378, 656, 117, 224, 548, 1456, 2912, 4934, 8868, 1771, 3543, ...
It's the powers of 2 written by skipping every other number. so 2, 4, 8, 16, 32, 64, 128...