Home > Polygonal number

But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):

Some numbers, like 36, can be arranged both as a square and as a triangle (see triangular square number):

The method for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as +.

Triangular numbers

1:

3:

6:

10:

Square numbers

1:

4:

9:

16:

Polygons with higher numbers of sides, such as pentagons and hexagons, can also be represented as arrangements of dots (by convention 1 is the first polygonal number for any number of sides).

Pentagonal numbers:

1:

5:

12:

22:

35:

Hexagonal numbers

1:

6:

15:

28:

45:

66: (which is also a triangular number and a sphenic number)

91:

If s is the number of sides in a polygon, the formula for the nth s-polygonal number is ½n((s-2)n - (s-4)).

The On-Line Encyclopedia of Integer SequencesThe On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. It is considered one of the major resources in mathematics. The database The Encyclopedia is a database recording information on integer sequence eschews terms using Greek prefixes (e.g., "octagonal") in favor of terms using numerals (i.e., "8-gonal").