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