Heptagonal number
Type of figurate number constructed by combining heptagons
title: "Heptagonal number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["figurate-numbers"] description: "Type of figurate number constructed by combining heptagons" topic_path: "general/figurate-numbers" source: "https://en.wikipedia.org/wiki/Heptagonal_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Type of figurate number constructed by combining heptagons ::
In mathematics, a heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The n-th heptagonal number is given by the formula :H_n=\frac{5n^2 - 3n}{2}. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ad/Heptagonal_numbers.svg" caption="The first five heptagonal numbers."] ::
The first few heptagonal numbers are: :0, 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, …
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
- Fib. Quart.]]'' '''43''' 3: 194
- "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers".
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