Heptagonal number

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, …

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

Additional properties

• The heptagonal numbers have several notable formulas: :H_{m+n}=H_m+H_n+5mn :H_{m-n}=H_m+H_n-5mn+3n :H_m-H_n=\frac{(5(m+n)-3)(m-n)}{2} :40H_n+9=(10n-3)^2

Sum of reciprocals

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

: \begin{align}\sum_{n=1}^\infty \frac{2}{n(5n-3)} &= \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5)+\frac{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)\ &=\frac13\left(\frac{\pi}{\sqrt[4]{5,\phi^6}}+\frac52\ln(5) -\sqrt5 \ln(\phi)\right)\ &=1.3227792531223888567\dots \end{align}

with golden ratio \phi = \tfrac{1+\sqrt5}2.

Heptagonal roots

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.

References

References

1. Fib. Quart.]]'' '''43''' 3: 194
2. "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers".