Heptagonal number

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 five heptagonal numbers.

The first few heptagonal numbers are:

1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … (sequence A000566 in the OEIS)

Parity

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.

Generalized heptagonal numbers

A generalized heptagonal number is obtained by the formula

$T_{n}+T_{\lfloor {\frac {n}{2}}\rfloor },$

where T_n is the nth triangular number. The first few generalized heptagonal numbers are:

1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, … (sequence A085787 in the OEIS)

Every other generalized heptagonal number is a regular heptagonal number. Besides 1 and 70, no generalized heptagonal numbers are also Pell numbers.

Sum of reciprocals

A formula for the sum of the reciprocals of the heptagonal numbers is given by:

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

Test for heptagonal numbers

${\sqrt {40n+9}}+3 \over 10$