A cool sequence problem…

Our oldest son (nearly 10) posed the following challenge:

What comes next in this list?

1, 1, 1, 2, 2, 3, 2, 4, 3, …

Answer and rationale (his and mine) after the jump…

He spoke the challenge from memory, and after having him repeat it twice, I typed it out (since there’s no paper handy).

After a bit, I tried putting it into rows and columns.  I start with 3 columns (since there were nine data):

1  1  1

2  2  3

2  4  3

but I saw nothing obvious.

Then I tried 2 columns, and hit paydirt:

1  1

1  2

2  3

2  4

3  ? …

And decided that the second column are the natural numbers, the first column are  ” (n+1)/2 rounded up”, in effect.

I told my son the next few terms in the sequence, and his eyes got big.

He was impressed that I “got it” so quickly, and I was impressed that he was thinking about zipper sequences.

I call a sequence a zipper sequence if it can be viewed as being built up by taking alternate terms from two simple sequences.  For example,    5, 2, 5, 2, 5, 2, …   is a simple zipper sequence built from two constant sequences.  A more complex example might be 1, 2, 3, 4, 9, 8, 27, 16, …

As you might expect, I was delighted by my son’s example.  “Where did you get this from?”, I asked.

“Oh, I made it up.”

So I asked that he tell me how he created it.  Here is what he said:

I just took three groups and put tallies in each of them. I put one on left side, one in middle, one on right, and so on.  I knew that by doing that pattern, the middle would have the most but I wanted to know how many it would have.

So he was thinking about putting tallies into three piles, passing from left to center to right to center to left to center to right to center etc…; his sequence was the count of how many tallies were in the “current” pile when a new tally was being placed.

After we compared methods, we both were excited.  He and I had totally different ways of thinking about 1, 1, 1, 2, 2, 3, 2, 4, 3, ….

Being in parent (not teacher) mode,  I didn’t push further to discuss  why his construction implied my formula.  But when I contemplated whether he and I should have that discussion, he spoke up and said he thought there must be other ways to think about this pattern as well.  “Maybe you should post this!”, he said.

We look forward to any other patterns or insights you might offer!