Type of figurate number constructed by combining heptagons
في الرياضيات ، العدد المسبع Heptagonal number هو عدد مضلعي يمثل شكل سباعي أضلاع . يعطى الرقم n منه بالعلاقة:
5
n
2
-
3
n
2
5
superscript
𝑛
2
3
𝑛
2
{\displaystyle{\displaystyle{\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
⌊
n
2
⌋
,
subscript
𝑇
𝑛
subscript
𝑇
𝑛
2
{\displaystyle{\displaystyle T_{n}+T_{\lfloor{\frac{n}{2}}\rfloor},}}
where T n n th 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 .[1]
خصائص إضافية The heptagonal numbers have several notable formulas:
H
m
+
n
=
H
m
+
H
n
+
5
m
n
subscript
𝐻
𝑚
𝑛
subscript
𝐻
𝑚
subscript
𝐻
𝑛
5
𝑚
𝑛
{\displaystyle{\displaystyle H_{m+n}=H_{m}+H_{n}+5mn}}
H
m
-
n
=
H
m
+
H
n
-
5
m
n
+
3
n
subscript
𝐻
𝑚
𝑛
subscript
𝐻
𝑚
subscript
𝐻
𝑛
5
𝑚
𝑛
3
𝑛
{\displaystyle{\displaystyle H_{m-n}=H_{m}+H_{n}-5mn+3n}}
H
m
-
H
n
=
(
5
(
m
+
n
)
-
3
)
(
m
-
n
)
2
subscript
𝐻
𝑚
subscript
𝐻
𝑛
5
𝑚
𝑛
3
𝑚
𝑛
2
{\displaystyle{\displaystyle H_{m}-H_{n}={\frac{(5(m+n)-3)(m-n)}{2}}}}
40
H
n
+
9
=
(
10
n
-
3
)
2
40
subscript
𝐻
𝑛
9
superscript
10
𝑛
3
2
{\displaystyle{\displaystyle 40H_{n}+9=(10n-3)^{2}}}
جمع المقلوبات A formula for the sum of the reciprocals of the heptagonal numbers is given by:[2]
∑
n
=
1
∞
2
n
(
5
n
-
3
)
=
1
15
π
25
-
10
5
+
2
3
ln
(
5
)
+
1
+
5
3
ln
(
1
2
10
-
2
5
)
+
1
-
5
3
ln
(
1
2
10
+
2
5
)
=
1
3
(
π
5
ϕ
6
4
+
5
2
ln
(
5
)
-
5
ln
(
ϕ
)
)
=
1.3227792531223888567
…
superscript
subscript
𝑛
1
2
𝑛
5
𝑛
3
absent
1
15
𝜋
25
10
5
2
3
5
1
5
3
1
2
10
2
5
1
5
3
1
2
10
2
5
missing-subexpression
absent
1
3
𝜋
4
5
superscript
italic-ϕ
6
5
2
5
5
italic-ϕ
missing-subexpression
absent
1.3227792531223888567
…
{\displaystyle{\displaystyle{\begin{aligned} \displaystyle\sum_{n=1}^{\infty}{%
\frac{2}{n(5n-3)}}&\displaystyle={\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)\\
&\displaystyle={\frac{1}{3}}\left({\frac{\pi}{\sqrt[4]{5\,\phi^{6}}}}+{\frac{5%
}{2}}\ln(5)-{\sqrt{5}}\ln(\phi)\right)\\
&\displaystyle=1.3227792531223888567\dots\end{aligned}}}}
with golden ratio
ϕ
=
1
+
5
2
italic-ϕ
1
5
2
{\displaystyle{\displaystyle\phi={\tfrac{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
=
40
x
+
9
+
3
10
,
𝑛
40
𝑥
9
3
10
{\displaystyle{\displaystyle n={\frac{{\sqrt{40x+9}}+3}{10}},}}
which is obtained by using the quadratic formula to solve
x
=
5
n
2
-
3
n
2
𝑥
5
superscript
𝑛
2
3
𝑛
2
{\displaystyle{\displaystyle x={\frac{5n^{2}-3n}{2}}}}
n .
انظر أيضاً المراجع
أعداد متعددات الحدود الأخرى
Possessing a specific set of other numbers
يمكن التعبير عنها بجموع معينة
