Stern's diatomic array is a certain array of integers.

The top row (row 0) consists of a single 1. Once we have constructed
the $n$th row, the next row is obtained by (1) beginning and ending the
row with a 1, (2) placing the sum of two adjacent numbers $i$ and $j$
directly between them in the row below (as in Pascal's triangle), and
(3) bringing down (copying) each number in row $n$ directly below in the
next row.

This array has a number of remarkable and unexpected properties. Here
are five problems in increasing order of difficulty.

1. The sum of the entries in row $n$ (beginning with row 0) is $3^n$.

2. The number of entries in row $n$ is $2^{n+1}-1$.

3. The alternating sum of the entries in row $n$ is $3^{n-1}$.

4. The largest entry in row $n$ is the $(n+1)$st Fibonacci number.

5. The sum of the cubes of the entries in row $n$ is $3 \times 7^{n-1}$.

We discuss these and further properties, including an amazing
bijection between positive integers and positive rational numbers.