# 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.

 $\displaystyle\begin{array}[]{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\ &&&&&&&&1&&1&&&&&&&\\ &&&&&&&1&&x&&1&&&&&&\\ &&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\ \end{array}$

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

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

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.

 $\displaystyle\begin{array}[]{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\ &&&&&&&&1&&1&&&&&&&\\ &&&&&&&1&&1&&1&&&&&&\\ &&&&&&1&&2&&2&&1&&&&&\\ &&&&&1&&2&&4&&2&&1&&&&\\ &&&&1&&3&&6&&6&&3&&1&&&\\ &&&1&&3&&9&&10&&9&&3&&1&&\\ &&1&&4&&12&&19&&19&&12&&4&&1&\\ &&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\ \end{array}$

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

## References

• 1 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
Title Losanitsch’s triangle LosanitschsTriangle 2013-03-22 15:44:09 2013-03-22 15:44:09 CompositeFan (12809) CompositeFan (12809) 13 CompositeFan (12809) Definition msc 05C38 Lozanic’s triangle Lozanić’s triangle