عدد مسبع

في الرياضيات، العدد المسبع Heptagonal number هو عدد مضلعي يمثل شكل سباعي أضلاع. يعطى الرقم n منه بالعلاقة:

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

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, … (المتتالية A000566 في 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.

 الأعداد المسبعة المعممة

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, … (المتتالية A085787 في OEIS)

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

 خصائص إضافية

• The heptagonal numbers have several notable formulas:

$H_{m+n}=H_m+H_n+5mn$

$H_{m-n}=H_m+H_n-5mn+3n$

$H_m-H_n=\frac{(5(m+n)-3)(m-n)}{2}$

$40H_n+9=(10n-3{)}^2$

 جمع المقلوبات

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

$\begin{array}{ll}\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n(5n-3)}& =\frac{1}{15}\pi \sqrt{25-10\sqrt{5}}+\frac{2}{3}\ln (5)+\frac{1+\sqrt{5}}{3}\ln \left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{1-\sqrt{5}}{3}\ln \left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)\\ & =\frac{1}{3}(\frac{\pi }{\sqrt[4]{5{\varphi }^6}}+\frac{5}{2}\ln (5)-\sqrt{5}\ln (\varphi ))\\ & =1.3227792531223888567\dots \end{array}$

with golden ratio $\varphi =\frac{1+\sqrt{5}}{2}$.

 الجذور المسبعة

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.

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