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Figurate number representing a decagon
In mathematics , a decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the n -th decagonal numbers counts the dots in a pattern of n nested decagons, all sharing a common corner, where the i th decagon in the pattern has sides made of i dots spaced one unit apart from each other. The n -th decagonal number is given by the following formula
d
n
=
4
n
2
−
3
n
{\displaystyle d_{n}=4n^{2}-3n}
.
The first few decagonal numbers are:
0 , 1 , 10 , 27 , 52 , 85 , 126 , 175 , 232 , 297 , 370, 451, 540, 637, 742, 855, 976, 1105 , 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000 , 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 (sequence A001107 in the OEIS ).
The n th decagonal number can also be calculated by adding the square of n to thrice the (n −1)th pronic number or, to put it algebraically, as
D
n
=
n
2
+
3
(
n
2
−
n
)
{\displaystyle D_{n}=n^{2}+3\left(n^{2}-n\right)}
.
Decagonal numbers consistently alternate parity .
D
n
{\displaystyle D_{n}}
is the sum of the first
n
{\displaystyle n}
natural numbers congruent to 1 mod 8.
D
n
{\displaystyle D_{n}}
is number of divisors of
48
n
−
1
{\displaystyle 48^{n-1}}
.
The only decagonal numbers that are square numbers are 0 and 1.
The decagonal numbers follow the following recurrence relations:
D
n
=
D
n
−
1
+
8
n
−
7
,
D
0
=
0
{\displaystyle D_{n}=D_{n-1}+8n-7,D_{0}=0}
D
n
=
2
D
n
−
1
−
D
n
−
2
+
8
,
D
0
=
0
,
D
1
=
1
{\displaystyle D_{n}=2D_{n-1}-D_{n-2}+8,D_{0}=0,D_{1}=1}
D
n
=
3
D
n
−
1
−
3
D
n
−
2
+
D
n
−
3
,
D
0
=
0
,
D
1
=
1
,
D
2
=
10
{\displaystyle D_{n}=3D_{n-1}-3D_{n-2}+D_{n-3},D_{0}=0,D_{1}=1,D_{2}=10}
The sum of the reciprocals of the decagonal numbers admits a simple closed form:
∑
n
=
1
∞
1
4
n
2
−
3
n
+
∑
n
=
1
∞
1
n
(
4
n
−
3
)
=
ln
(
2
)
+
π
6
.
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{4n^{2}-3n}}+\sum _{n=1}^{\infty }{\frac {1}{n\left(4n-3\right)}}=\ln \left(2\right)+{\frac {\pi }{6}}.}
This derivation rests upon the method of adding a "constructive zero":
∑
n
=
1
∞
1
n
(
4
n
−
3
)
=
4
3
∑
n
=
1
∞
(
1
4
n
−
3
−
1
4
n
)
=
2
3
∑
n
=
1
∞
(
2
4
n
−
3
−
2
4
n
+
(
1
4
n
−
1
−
1
4
n
−
2
)
−
(
1
4
n
−
1
−
1
4
n
−
2
)
)
{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n\left(4n-3\right)}}&{}={\frac {4}{3}}\sum _{n=1}^{\infty }\left({\frac {1}{4n-3}}-{\frac {1}{4n}}\right)\\&={\frac {2}{3}}\sum _{n=1}^{\infty }\left({\frac {2}{4n-3}}-{\frac {2}{4n}}+\left({\frac {1}{4n-1}}-{\frac {1}{4n-2}}\right)-\left({\frac {1}{4n-1}}-{\frac {1}{4n-2}}\right)\right)\end{aligned}}}
Rearranging and considering the individual sums:
=
2
3
∑
n
=
1
∞
[
(
1
4
n
−
3
−
1
4
n
−
2
+
1
4
n
−
1
−
1
4
n
)
+
(
1
4
n
−
2
−
1
4
n
)
+
(
1
4
n
−
3
−
1
4
n
−
1
)
]
=
2
3
∑
n
=
1
∞
(
1
4
n
−
3
−
1
4
n
−
2
+
1
4
n
−
1
−
1
4
n
)
+
1
3
∑
n
=
1
∞
(
1
2
n
−
1
−
1
2
n
)
+
2
3
∑
n
=
1
∞
(
1
2
(
2
n
−
1
)
−
1
−
1
2
(
2
n
)
−
1
)
=
2
3
∑
n
=
1
∞
(
−
1
)
n
+
1
n
+
1
3
∑
n
=
1
∞
(
−
1
)
n
+
1
n
+
2
3
∑
n
=
1
∞
(
−
1
)
n
+
1
2
n
−
1
=
ln
(
2
)
+
π
6
.
{\displaystyle {\begin{aligned}&={\frac {2}{3}}\sum _{n=1}^{\infty }\left[\left({\frac {1}{4n-3}}-{\frac {1}{4n-2}}+{\frac {1}{4n-1}}-{\frac {1}{4n}}\right)+\left({\frac {1}{4n-2}}-{\frac {1}{4n}}\right)+\left({\frac {1}{4n-3}}-{\frac {1}{4n-1}}\right)\right]\\&={\frac {2}{3}}\sum _{n=1}^{\infty }\left({\frac {1}{4n-3}}-{\frac {1}{4n-2}}+{\frac {1}{4n-1}}-{\frac {1}{4n}}\right)+{\frac {1}{3}}\sum _{n=1}^{\infty }\left({\frac {1}{2n-1}}-{\frac {1}{2n}}\right)+{\frac {2}{3}}\sum _{n=1}^{\infty }\left({\frac {1}{2(2n-1)-1}}-{\frac {1}{2(2n)-1}}\right)\\&={\frac {2}{3}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}+{\frac {1}{3}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}+{\frac {2}{3}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n-1}}\\&=\ln \left(2\right)+{\frac {\pi }{6}}.\end{aligned}}}
Possessing a specific set of other numbers
Expressible via specific sums