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
=
5
n
2
−
3
n
2
{\displaystyle H_{n}={\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, … (sequence A000566 in the 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 .
Sum of reciprocals
A formula for the sum of the reciprocals of the heptagonal numbers is given by:[1]
∑
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
…
{\displaystyle {\begin{aligned}\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)\\&={\frac {1}{3))\left({\frac {\pi }{\sqrt[{4}]{5\,\phi ^{6))))+{\frac {5}{2))\ln(5)-{\sqrt {5))\ln(\phi )\right)\\&=1.3227792531223888567\dots \end{aligned))}
with golden ratio
ϕ
=
1
+
5
2
{\displaystyle \phi ={\tfrac {1+{\sqrt {5))}{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
=
40
x
+
9
+
3
10
,
{\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
{\displaystyle x={\frac {5n^{2}-3n}{2))}
for its unique positive root n .