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Number Chu Space Triangle Similarity In Geometry Binomial Coefficient Graph Theory Pascals Triangle

1	Triangle
1	Similarity In Geometry
1	Chu Space
1	Number
2	Binomial Coefficient
3	Graph Theory
6	Pascals Triangle

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Losanitschs 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 numbered 0, and the 1's position as 0.

$\begin{displaymath}\begin{array}{cccccccccccccccccc} & & & & & & & & & 1 & & & &... ... & &\vdots & & & & \vdots & & & & \vdots& & & & \\ \end{array}\end{displaymath}$

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

$\displaystyle {{\frac{n}{2}-1} \choose {{k - 1} \over 2}}$

From this point 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.

$\begin{displaymath}\begin{array}{cccccccccccccccccc} & & & & & & & & & 1 & & & &... ... & &\vdots & & & & \vdots & & & & \vdots& & & & \\ \end{array}\end{displaymath}$

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

Bibliography

1: S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.