Heptagonal Pyramidal Numbers which are Square

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 $\displaystyle$ $=$ $\displaystyle 196$

$\blacksquare$