Polygonal number

Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of oblong, triangular, and square numbers[1]: 1 .

Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above.

A property of this table can be expressed by the following identity (see A086270): with Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets.

The problem of finding numbers that belong to three polygonal sets is more difficult.

[5] The number 1225 is hecatonicositetragonal (s = 124), hexacontagonal (s = 60), icosienneagonal (s = 29), hexagonal, square, and triangular.