Definition:Polygonal Number
Definition
Polygonal numbers are those denumerating a collection of objects which can be arranged in the form of an regular polygon.
A polygonal number is an integer defined recursively as follows:
- $\forall k \in \Z_{\ge 2}: \forall n \in Z_{\ge 0}: \map P {k, n} = \begin{cases}
0 & : n = 0 \\ \map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0 \end{cases}$
For a given $k$, polygonal numbers are referred to by the name of the appropriate $k$-sided polygon.
For large $k$, they are therefore called (when used) $k$-gonal numbers.
Also known as
When referring to a $k$-gonal number where $k$ is a more complex expression than just a single number or letter, it may be less unwieldy to refer to it as a polygonal number of order $k$.
Examples
Triangular Numbers
When $k = 3$, the recurrence relation is:
- $\forall n \in \N: T_n = \map P {3, n} = \begin{cases}
0 & : n = 0 \\ \map P {3, n - 1} + \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
See Triangular Number.
Square Numbers
When $k = 4$, the recurrence relation is:
- $\forall n \in \N: S_n = \map P {4, n} = \begin{cases}
0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
See Square Number.
Square numbers are of course better known as:
- $S_n = n^2$
Pentagonal Numbers
When $k = 5$, the recurrence relation is:
- $\forall n \in \N: P_n = \map P {5, n} = \begin {cases}
0 & : n = 0 \\ \map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0 \end {cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
See Pentagonal Number.
Hexagonal Numbers
When $k = 6$, the recurrence relation is:
- $\forall n \in \N: H_n = \map P {6, n} = \begin{cases}
0 & : n = 0 \\ \map P {6, n - 1} + 4 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
See Hexagonal Number.
Heptagonal Numbers
When $k = 7$, the recurrence relation is:
- $\forall n \in \N: H_n = \map P {7, n} = \begin{cases}
0 & : n = 0 \\ \map P {7, n - 1} + 5 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
See Heptagonal Number.
Octagonal Numbers
When $k = 8$, the recurrence relation is:
- $\forall n \in \N: O_n = \map P {8, n} = \begin{cases}
0 & : n = 0 \\ \map P {8, n - 1} + 6 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
See Octagonal Number.
Degenerate Case
Consider the polygonal number $P \left({2, n}\right)$ when $k = 2$.
In this case, the polygon degenerates into a straight line, and the recurrence formula becomes:
- $P \left({2, n}\right) = \begin{cases}
0 & : n = 0 \\ P \left({2, n-1}\right) + 0 \times \left({n-1}\right) + 1 & : n > 0 \end{cases}$
Hence:
- $P \left({2, n}\right) = P \left({2, n-1}\right) + 1$
and the sequence goes:
- $0, 1, 2, 3, \ldots$
which is of course the natural numbers.
Also see
Historical Note
Polygonal numbers were originally studied by the ancient Greeks.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$