The terms increase if and only if 2a + 3b > 10a + b, that is iff b > 4a. As b is a single digit number the only terms that are followed by larger terms are 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 18, 19 and 29. For all other values the sequence is decreasing.

Large terms reduce by one fifth in order of magnitude, and once the sequence reaches 2 digit numbers the largest value is given by a=b=9 which maps to 45. Numbers less than 45 map to 33 or less because 2a+3b < 33. After that there are a finite number of values the sequence can take so values are repeated and cycles occur.

The only 2-cycle is 7, 21.
The 6-cycles are:
2,6,18,26,22,10,2...
4,12,8,24,16,20,4...
5,15,17,23,13,11,5...
9,27,25,19,29,31,9...
par Other terms which are not in the cycles but lead into the cycles are:
1,3, ® 9 etc.
30 ® 6 etc.
32 ® 12 etc.
33 ® 15 etc.

We have proved that all higher numbers reduce to 33 or less so sequences with all other starting points enter one of the listed cycles or end up at a fixed point.