Editing Heptagonal number

{{Short description|Type of figurate number constructed by combining heptagons}}
In [[mathematics]], a '''heptagonal number''' is a [[figurate number]] that is constructed by combining [[heptagon]]s with ascending size. The ''n''-th heptagonal number is given by the formula 
:<math>H_n=\frac{5n^2 - 3n}{2}</math>.
[[File:Heptagonal numbers.svg|thumbnail|right|The first five heptagonal numbers.]]
The first few heptagonal numbers are:
:[[0 (number)|0]], [[1 (number)|1]], [[7 (number)|7]], [[18 (number)|18]], [[34 (number)|34]], [[55 (number)|55]], [[81 (number)|81]], [[112 (number)|112]], [[148 (number)|148]], [[189 (number)|189]], [[235 (number)|235]], 286, 342, 403, 469, 540, [[616 (number)|616]], 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … {{OEIS|id=A000566}}

==Parity==
The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like [[square number]]s, 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]].
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==Generalized heptagonal numbers==
A '''generalized heptagonal number''' is obtained by the formula
:<math>T_n + T_{\lfloor \frac{n}{2} \rfloor},</math>
where ''T''<sub>''n''</sub> is the ''n''th triangular number. The first few generalized heptagonal numbers are:
:1, [[4 (number)|4]], 7, [[13 (number)|13]], 18, [[27 (number)|27]], 34, [[46 (number)|46]], 55, [[70 (number)|70]], 81, [[99 (number)|99]], 112, … {{OEIS|id=A085787}}

Every other generalized heptagonal number is a regular heptagonal number. Besides 1 and 70, no generalized heptagonal numbers are also [[Pell number]]s.<ref>B. Srinivasa Rao, "Heptagonal Numbers in the Pell Sequence and [[Diophantine equation]]s <math>2x^2 = y^2(5y - 3)^2 \pm 2</math>" ''[[Fibonacci Quarterly|Fib. Quart.]]'' '''43''' 3: 194</ref>
-->
==Additional properties==
* The heptagonal numbers have several notable formulas:
:<math>H_{m+n}=H_m+H_n+5mn</math>
:<math>H_{m-n}=H_m+H_n-5mn+3n</math>
:<math>H_m-H_n=\frac{(5(m+n)-3)(m-n)}{2}</math>
:<math>40H_n+9=(10n-3)^2</math>
==Sum of reciprocals==
A formula for the [[sums of reciprocals|sum of the reciprocals]] of the heptagonal numbers is given by:<ref>{{Cite web |url=http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |title=Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers |access-date=2010-05-19 |archive-date=2013-05-29 |archive-url=https://web.archive.org/web/20130529032918/http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |url-status=dead }}</ref>

:<math>
\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{{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)\\
&=\frac13\left(\frac{\pi}{\sqrt[4]{5\,\phi^6}}+\frac52\ln(5) -\sqrt5 \ln(\phi)\right)\\
&=1.3227792531223888567\dots
\end{align}    
</math>

with [[golden ratio]] <math>\phi = \tfrac{1+\sqrt5}2</math>.

== 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

:<math>n = \frac{\sqrt{40x + 9} + 3}{10},</math>

which is obtained by using the [[quadratic formula]] to solve <math>x = \frac{5n^2 - 3n}{2}</math> for its unique positive root ''n''.

==References==

<references/>


{{Figurate numbers}}
{{Classes of natural numbers}}
{{series (mathematics)}}

[[Category:Figurate numbers]]