* In this case, "non-trivial" means not a simple arithmetic (n, n+1, n+2, n+3...) or geometric (n×1, n×2, n×3...) sequence.
My first inclination was to go with the primes (2, 3, 5, 7, 11...). Though if we allow rounding (like a ceiling function), ceil[primes/2] might work (1, 2, 3, 4, 6...).
What do you think? An OEIS search
has not turned up anything. I'm sure someone's looked into this before, but as much as my modest research has stumped me, I'm just as interested in the thought process/derivation of the answer. There is probably a really elegant answer I'm missing!
As far as the practical application: Well... say, hypothetically, that one had a password that was required to change every so often (and not change back), and one wanted to keep things simple by just changing (in a not-easily-guessable way) a whole number appended to the end of one's password (one that won't blow up to, say... 80 digits by step #20 or so). [I will not use this method, nor should you. Probably.]