Closed Form for Polygonal Numbers/Examples
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Examples of Closed Form for Polygonal Numbers
Let $\map P {k, n}$ be the $n$th $k$-gonal number.
The closed-form expression for $\map P {k, n}$ for various $k$ can be expressed as:
\(\ds k = 3: \ \ \) | \(\ds \frac n 2 \paren {\paren {3 - 2} n - 3 + 4}\) | \(=\) | \(\ds \frac n 2 \paren {n + 1}\) | \(\ds \quad = \frac {n \paren {n + 1} } 2\) | Definition of Triangular Number | |||||||||
\(\ds k = 4: \ \ \) | \(\ds \frac n 2 \paren {\paren {4 - 2} n - 4 + 4}\) | \(=\) | \(\ds \frac n 2 \paren {2 n - 0}\) | \(\ds \quad = n^2\) | Definition of Square Number | |||||||||
\(\ds k = 5: \ \ \) | \(\ds \frac n 2 \paren {\paren {5 - 2} n - 5 + 4}\) | \(=\) | \(\ds \frac n 2 \paren {3 n - 1}\) | \(\ds \quad = \frac {n \paren {3 n - 1} } 2\) | Definition of Pentagonal Number | |||||||||
\(\ds k = 6: \ \ \) | \(\ds \frac n 2 \paren {\paren {6 - 2} n - 6 + 4}\) | \(=\) | \(\ds \frac n 2 \paren {4 n - 2}\) | \(\ds \quad = n \paren {2 n - 1}\) | Definition of Hexagonal Number | |||||||||
\(\ds k = 7: \ \ \) | \(\ds \frac n 2 \paren {\paren {7 - 2} n - 7 + 4}\) | \(=\) | \(\ds \frac n 2 \paren {5 n - 3}\) | \(\ds \quad = \frac {n \paren {5 n - 3} } 2\) | Definition of Heptagonal Number | |||||||||
\(\ds k = 8: \ \ \) | \(\ds \frac n 2 \paren {\paren {8 - 2} n - 8 + 4}\) | \(=\) | \(\ds \frac n 2 \paren {6 n - 4}\) | \(\ds \quad = n \paren {3 n - 2}\) | Definition of Octagonal Number |
and so on.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $45$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $45$