If you look at the number 45678910 you can see a clear linear pattern of growth by a factor of one, and if there were blanks for you to fill in you’d be able to do so to keep the pattern consistent. However, what’s rare about this number (I don’t know how rare) is that if you switch the two last digits, the “1” with the “0”, to get 45678901, you don’t have a broken pattern—you switch to a totally new pattern. What’s more, it’s a qualitatively distinct pattern, as you are moving from a linear pattern to a cyclical pattern (45678901234567890123…). Are there any other patterns out there that you can switch the final two digits in an already well established pattern and get a new pattern, not just a broken sequence?
I wouldn’t spend too much time trying to think of one deductively, though, as there is too much living to get stuck in math puzzles! This was just a situation of inductive luck, where luck was the collision of drifting thoughts and awareness.
Note: (April 19, 2015) - The pattern is made longer by changing the numeral system from decimal (ten single digits) to hexadecimal (sixteen single digits), for example. Further, I started at the #4 for a curious reason to perhaps be elucidated in the future, but one could have started at the #1 or 0, or possibly even a negative number, though that might have broken the pattern rules somewhat... though I'm no referee of the math jungle, just an occasional adventurer there.