Integer Sequences
I'm becoming interested in integer sequences. An integer sequence is simply a list of integers (which are "whole numbers," i.e. numbers without decimal places). So for example, 1 5 8 135234 30 1 is a sequence that has no particular significance to me (I just made it up) and 1 2 3 4 5 6... is another sequence that is pretty important to us all. Some sequences are finite (like that first one) and some never end (like the second one).
There's a website (The On-Line Encyclopedia of Integer Sequences) that catalogs integer sequences. Obviously, there are an infinite number of sequences (every list of numbers is a new sequence). But, some sequences are more important than the others. The sequence 1 2 3 4... is very important, and is the first one that children learn. The sequence 2 4 6 8... (i.e. all even numbers in numerical order) is also important. Others are less obvious, but still important: 8, 5, 4, 9, 1, 7, 6, 3, 2, 0 (this is the order the numbers would appear if you spelled out the numbers' names and sorted them alphabetically).
So clearly I'm using the word "important" loosely. Some sequences are undeniably important, such as the famous Fibonacci sequence, while others are merely interesting or pretty or nifty or clever. Maybe some don't seem important now, but you'll never know when somebody might need them.
To date I've only contributed one sequence the online encyclopedia. It's pretty stupid, but I wanted to start easy.
I'm currently working on adding a term to a sequence, though. See, I thought I had thought up a with a sequence to add to the encyclopedia, but it turned out someone had already thought of it. However, that person included only 7 numbers of the sequence (it's an infinite sequence). I think I can add the 8th term.
The sequence is pretty easy to explain: The number pi is a never-ending string of digits (3 1 4 1 5 9...). The sequence I'm working with uses these digits as a starting point. The idea is to start at the beginning of pi, and move along to the right until you visited all the single-digit numbers at least once. Turns out you have to go 32 steps before you hit all the numbers 0-9. So mark 32 as your first term in the sequence. Then do the same process but look for all the two digit numbers (00-99) as you group the digits of pi in twos (e.g. 31 14 41 15 59...You have to go 606 places before you find them, thus 606 is the next term. Continue on, looking for all three digit numbers, four digit numbers, etc.
The sequence thus far in the encyclopedia is 32 606 8555 99849 1369564 14118312 166100506. So they know how far you have to scan to find all the numbers up to 9,999,999 (technically we're not looking for all the numbers since 00 is the same number as 0. What we're looking for is more accurately called a string). You have to ask a computer to do the boring work for you, of course, but at around the 7th term, you have to be careful about using computer memory efficiently and making a fast program, because even a computer has trouble keeping track of that many numbers. It's right there on the edge, but I think I've made my program fast enough and efficient enough to get the 8th term.
My current problem is getting enough digits of pi.
