Closed Form for Polygonal Numbers

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Theorem

Let $\map P {k, n}$ be the $n$th $k$-gonal number.


The closed-form expression for $\map P {k, n}$ is given by:

$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$


Proof

By definition of the $n$th $k$-gonal number:

$\map P {k, n} = \begin{cases} 0 & : n = 0 \\ \map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0 \end{cases}$


Then:

$\paren {\paren {k - 2} \paren {j - 1} + 1}$

is an arithmetic progression.

Its initial term $a$ is $1$, and its common difference $d$ is $k - 2$.


Hence:

\(\displaystyle \map P {k, n}\) \(=\) \(\displaystyle \sum_{j \mathop = 1}^n \paren {\paren {k - 2} \paren {j - 1} + 1}\) Sum of Arithmetic Progression
\(\displaystyle \) \(=\) \(\displaystyle \frac {n \paren {2 + \paren {n - 1} \paren {k - 2} } } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \frac n 2 \paren {\paren {k - 2} n - \paren {k - 2} + 2}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac n 2 \paren {\paren {k - 2} n - k + 4}\)

as required.

$\blacksquare$


Examples

The closed-form expression for $\map P {k, n}$ for various $k$ can be expressed as:

\(\displaystyle k = 3: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {3 - 2} n - 3 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {n + 1}\) \(\displaystyle \quad = \frac {n \paren {n + 1} } 2\) Definition of Triangular Number
\(\displaystyle k = 4: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {4 - 2} n - 4 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {2 n - 0}\) \(\displaystyle \quad = n^2\) Definition of Square Number
\(\displaystyle k = 5: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {5 - 2} n - 5 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {3 n - 1}\) \(\displaystyle \quad = \frac {n \paren {3 n - 1} } 2\) Definition of Pentagonal Number
\(\displaystyle k = 6: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {6 - 2} n - 6 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {4 n - 2}\) \(\displaystyle \quad = n \paren {2 n - 1}\) Definition of Hexagonal Number
\(\displaystyle k = 7: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {7 - 2} n - 7 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {5 n - 3}\) \(\displaystyle \quad = \frac {n \paren {5 n - 3} } 2\) Definition of Heptagonal Number
\(\displaystyle k = 8: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {8 - 2} n - 8 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {6 n - 4}\) \(\displaystyle \quad = n \paren {3 n - 2}\) Definition of Octagonal Number

and so on.


Sources