Losanitsch’s triangle

A triangular arrangement of numbers very similar to Pascal’s triangle.

Begin as you would if you were constructing Pascal’s triangle, with a 1 in the top row, and that row $k$ numbered 0, and the 1’s position $n$ as 0.

Now, for the next value, add up the two values above, but then subtract

$$\left(\genfrac{}{}{0pt}{}{\frac{n}{2}-1}{\frac{k-1}{2}}\right)$$

From this forward, do the same for every even-numbered position in an even-numbered row. Instead of calculating the binomial coefficient, it can be looked up in Pascal’s triangle.

This triangle was first studied by the Serbian chemist Sima Losanitsch, but has since been found to have applications in graph theory and combinatorics.