The first sequence is more fun than interesting.

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,

Do you see the pattern? This is a tough one and most people won't see it. Try again, this time, reading the values out loud. One, One-One, Two-One, One-Two One-One, One-One One-Two Two-One, etc...

See it yet?

The second sequence is far more fascinating.

1, 2, 4, 8, 16, 22,

**26, 38, 62, 74, 102**, 104, 108, 116, 122, 126, 138, 162, 174, 202, 206, 218, 234, 258, 338, 410, 414, 430, 442, 474, 586, 826, 922, 958, 1318, 1342, 1366, ...

This one is a bit more simple, yet it contains some interesting properties. I'm not great at speaking in math terms, but I'll give it a try: this sequence is created by: given a value, x. You multiply all the non-zero digits of x together and then add that value to x. Thus with a value like 62, you multiply 6 x 2 and get 12. 62 + 12 = 74, which is the next value in the sequence.

Who cares, right? It's really not THAT different from the Look/Say, after all, it's just playing with the digits of values, big deal, right? Well, notice this:

3, 6, 12, 14, 18,

**26, 38, 62, 74, 102**, ...

We started from a different seed, but the two sequences joined at 26. The man whose done a lot of research on this is named Paul Loomis, he theorizes that starting from any positive integer, its sequence will join the sequence started from 1. A C program he wrote proved this to be true for all values less than 1 million. Interestingly, this behavior can be seen for all bases, not just base 10. I actually contacted Paul about this sequence, I was curious what he called it, he told me they're called Digit Product Sequences. Expect a snippet that not only models this sequence, but also has a built-in base changer, coming soon

I started looking for other sequences, but found it hard to find any others that had properties that were truly interesting. So, after a nice, long build-up, that leads me to my question. Do you guys know of any other awesome sequences? If so, I'd love to discuss them.

This post has been edited by **atraub**: 07 June 2012 - 12:18 AM