I've been mulling over some sequences that are rather beautiful for their surprising properties and even more fascinating applications. The point of this "contest" is just to show off a sequence that you really love and that you think isn't that very well known. The person whose post receives the most number of votes will be declared the person with the most interesting sequence in the world! A post should explain why the sequence is interesting or at least include a fun or exciting property.
To kick things off, I'm going to share with you my submission for the most interesting sequence in the world, the Thue–Morse sequence. It goes like this:
Interesting Property: Let's have a look at Note that the 1st, 4th, 6th, and 7th positions are marked with a "0" while the 2nd, 3rd, 5th, and 8th positions are marked with a "1". Now examine the following equality:
Whoa, is this just coincidence? Nope! It turns out that this pattern generalizes for later terms and for higher powers than Let denote the digit in the th position of It turns out that the sum of the th powers of those for which is equal to the sum of the th powers of all of the 's for which
So with that in mind, because we know that
Pretty interesting, right? I know there are plenty of other fun sequences out there to be shared and probably much more interesting than this one, so go hunt them down! Anyway, our usual Problem Writing Party will resume next week while I put together the new challenge quizzes that the community has created.
Good luck finding the most interesting sequence. I'm excited to see what the community can come up with.