Mathematical curiosities: pentagonal identities. (English)

Parabola 51, No. 1, 3 p., electronic only (2015).

From the text: Polygonal numbers enumerate the number of points in a regular geometrical arrangement of the points in the shape of a regular polygon. An example is the triangular number $T_n$ which enumerates the number of points in a regular triangular lattice of points whose overall shape is a triangle. Square numbers $S_n$ can be defined in a similar fashion enumerating the number of points in a regular square lattice of points whose overall shape is a square. The geometrical construction for the first four pentagonal numbers is shown in a figure. Hexagonal numbers, heptagonal numbers etc. can also be defined. In general if $s$ denotes the number of sides of a regular polygon then the $n$th $s$-gonal number is given by $θ(s,n)=\frac{1}{2}(n^2(s-2)-n(s-4))$. A curiosity considered here is that the sum of two polygonal numbers $θ(s,n)$ and $θ(s,m)$ is sometimes equal to the number composed of the concatenation of $n$ and $m$. For example, among the pentagonal numbers, $P_3=12$ and $P_4=22$, so that $P_3+P_4=34$, which is the concatenation of 3 and 4. We will refer to such pairs as polycat pairs.