1, 2, 4, 8, 16, …

or

A, B, AA, BB, AAA, BBB, ….

In school mathematics, sequences are often called "patterns." I don't like that usage, because it's terribly limiting. Mathematical patterns can be found in visual designs, crystals, the multiplication table, or some totality of facts…patterns aren't just about sequences! You'll often hear people say, "Mathematics is the study of patterns." They don't mean, "Mathematics is the study of sequences."

My colleague William McCallum has a useful dictum: "Patterns are a tool, not a topic." For example, patterns in the multiplication table could be used as a tool for teaching about the properties of operations; patterns in the sequence 1, 2, 4, 8, 16 could be used as a tool for teaching about exponential functions. Whatever work is done with sequences at a given grade ought to transcend 'patternology' to intentionally build up students' strengths in the most important mathematical topics at each grade level.

***

Lately in idle moments I'll try to pass the time by thinking up a sequence of numbers that isn't yet included in the Online Encyclopedia of Integer Sequences (OEIS). It's a fun game—like trying to think of a notable topic that isn't in Wikipedia.

This was my first attempt:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 30, 41, 50, 61, 70, 81, 90, 111, 200, ...

The rule of this sequence is that, beginning with zero, each successive number must share no digits with the previous number—and be as small as possible subject to that constraint. For example, the number after 200 will be 311, because 311 is the smallest number greater than 200 that doesn't have 2 or 0 as a digit.

This is sequence A030283 in OEIS.

To create the sequence whose initial terms are in the title of this post, I considered a sequence of regular polygons with sides of unit length (triangle, square, pentagon, hexagon, etc.). Choosing a standard position for the polygons (centered at the origin), and also choosing a standard orientation (with the top edge horizontal), I counted how many grid points (

*m*,

*n*) were inside the polygon or on its boundary. (Here

*m*and

*n*are integers.)

Here are some diagrams showing the first eleven values 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13:

My sequence doesn't appear to be in OEIS. There's a partial match with A219844, but the two sequences aren't the same.

***

When you look at the numbers 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, what do you notice? Will the feature you noticed hold true for the entire sequence, out to infinity?

Feel free to write any observations, conjectures or arguments down in the comments. Also, feel free to put a sequence of your own in the comments!

***

My daughter and I talked a bit about sequences during Saturday School a few weeks ago. That morning, my daughter was writing in a workbook she'd brought home from school. Looking across the table, I noticed that whoever had written the workbook had misunderstood 4.OA.5, a standard that involves sequences. In an effort to rescue the math, I quickly sketched the following problem:

*Start with 256. At each step, divide by 2. Repeat forever!*

*a) Show 4 steps of the pattern (sequence)*

*b) Prove or disprove: every number in this pattern (sequence) is a whole number.*

(As you can see, I took this opportunity to reinforce fractions and calculation of quotients. Also, note that the rule for the sequence is

*given*. This is a math problem, not an IQ test.)

*all*of the numbers in the sequence would be whole numbers. That gambit didn't work, however—she saw from the outset that the values would eventually drop below 1, and she proceeded to show this by generating sufficiently many values. (Actually, more than enough values! I then drew her attention to the second of the two arguments below.)

My daughter had fun solving this, and we enjoyed talking about it. What's cooler than infinity? Done right, I think 4.OA.5 can prompt some very good mathematics. I expect it has also helped publishers and school systems hang onto old-style, non-aligned "pattern" work in the curriculum.