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