# Heptagonal Pyramidal Numbers which are Square

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## Theorem

The sequence of heptagonal pyramidal numbers which also have the property of being square begins:

- $0, 1, 196, \ldots$

## Proof

\(\displaystyle \) | \(\) | \(\displaystyle \dfrac {0 \left({0 + 1}\right) \left({5 \times 0 - 2}\right)} 6\) | Closed Form for Heptagonal Pyramidal Numbers | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {0 \times 1 \times \left({- 2}\right)} 6\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0^2\) | Definition of Square Number |

\(\displaystyle \) | \(\) | \(\displaystyle \dfrac {1 \left({1 + 1}\right) \left({5 \times 1 - 2}\right)} 6\) | Closed Form for Heptagonal Pyramidal Numbers | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {1 \times 2 \times 3} 6\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1^2\) | Definition of Square Number |

\(\displaystyle \) | \(\) | \(\displaystyle \dfrac {6 \left({6 + 1}\right) \left({5 \times 6 - 2}\right)} 6\) | Closed Form for Heptagonal Pyramidal Numbers | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {6 \times 7 \times 28} 6\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 7 \times \left({2^2 \times 7}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({2 \times 7}\right)^2\) | Definition of Square Number | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 196\) |

$\blacksquare$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $196$