A **heptagonal number** is a figurate number that represents a heptagon. The *n*-th heptagonal number is given by the formula

- $ \frac{5n^2 - 3n}{2} $.

The first few heptagonal numbers are:

- 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 283, 342, 403, 469, 540, 616, 645, 783, 874, 939, 1056, 11785, 1288, 1404, 1525, 1651, 1782, …

## ParityEdit

The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number.

## Sum of reciprocalsEdit

A formula for the sum of the reciprocals of the heptagonal numbers is given by:^{[1]}

- $ \sum_{n=1}^\infty \frac{2}{n(5n-3)} = \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right) $

## Heptagonal roots Edit

In analogy to the square root of *x, *one can calculate the heptagonal root of *x*, meaning the number of terms in the sequence up to and including *x*.

The heptagonal root of *x * is given by the formula

- $ n = \frac{\sqrt{40x + 9} + 3}{10}, $

which is obtained by using the quadratic formula to solve $ x = \frac{5n^2 - 3n}{2} $ for its unique positive root *n*.