It's always nice to have a special example to get students interested in the material. If you're teaching sequences, the Recaman sequence is one of those examples that your students will remember long after they graduate. I first learned about it when I read the book "Here's Looking at Euclid" by Alex Bellos. Here's the beginning of the Recaman Sequence: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155...

Do you see any pattern to this sequence? I sure didn't, but it's actually created by a simple rule: $latex a_n=a_{n-1}-n$ if $latex a_n>0$ and has not appeared in the sequence yet and $latex a_n=a_{n-1}+n$ otherwise. Start with $latex a_0=0$. Since $latex a_1=a_0-1<0$, the first term of the sequence is $latex a_1=a_0+1=1$. To get the second term of the sequence, notice that $latex a_1-2=1-2<0$, so $latex a_2=a_1+2=1+2=3$. For the third term of the sequence, $latex a_2-3=3-3=0$, which is in the sequence already, so $latex a_3=a_2+3=3+3=6$. Continuing the process results in the sequence above. So you have an orderly rule that gives rise to a sequence that doesn't look particularly orderly. If you graph the sequence, however, the order emerges in stunning fashion. Here's the output I got with the help of Sage:

That pattern can also be heard in the music created by the sequence: according to the Alex Bellos' book "Here's Looking at Euclid", the notes of the music are determined by the 88 keys of the piano where the first key of the piano matches up with the first term (1) of the sequence. The second term (3) would be the third key of the keyboard and so on until a term is bigger than 88, at which point you continue back at the bottom of the keys (the term 89 would be the first key of the piano). To hear the music of the sequence, go here. You can pick other features of the music and when you're read, press "PLAY". For me, nothing played; instead it downloaded a file to my computer which I had to open on my own. The result? Music that definitely had order to it, something you might never guess by looking at the sequence. And a musical aspect to sequences might just help motivate some of your students, too.

I've placed the Sage code to create a manipulative for graphing the Recaman sequence. You can find it on the Python/Sage page and the screenshot of the output is on the Sage Output page.