mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a decagonal number is a

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...

that extends the concept of

triangular A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

and

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

s to the

decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. Regular decagon A '' regular decagon'' has a ...

(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 = 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 Year 1105 ( MCV) was a common year starting on Sunday of the Julian calendar. Events By place Levant * February 28 – Raymond IV dies at his castle of Mons Peregrinus ("Pilgrim's Mountain") near Tripoli. Raymond leaves his 2- ...

, 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 . The ''n''th decagonal number can also be calculated by adding the square of ''n'' to thrice the (''n''−1)th

pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...

or, to put it algebraically, as :

D_n = n^2 + 3\left(n^2 - n\right)

Properties

* Decagonal numbers consistently alternate parity. *

D_n

is the sum of the first

n

natural numbers congruent to 1 mod 8. *

D_n

is number of divisors of

48^

. * The only decagonal numbers that are square numbers are 0 and 1. * The decagonal numbers follow the following recurrence relations: :

D_n=D_+8n-7 , D_0=0

D_n=2D_-D_+8, D_0=0,D_1=1

D_n=3D_-3D_+D_, D_0=0, D_1=1, D_2=10

Sum of reciprocals

The sum of the reciprocals of the decagonal numbers admits a simple closed form:

\sum_^\frac+\sum_^\frac=\ln\left(2\right)+\frac.

Proof

This derivation rests upon the method of adding a "constructive zero":

\begin
\sum_^\frac &  =\frac\sum_^\left(\frac-\frac\right) \\
&=\frac\sum_^\left(\frac-\frac+\left(\frac-\frac\right)-\left(\frac-\frac\right)\right)
\end

Rearranging and considering the individual sums:

\\ &= \frac \sum_^ \left( \frac - \frac + \frac - \frac \right) + \frac \sum_^ \left( \frac - \frac \right) + \frac \sum_^ \left( \frac - \frac \right) \\ &= \frac \sum_^ \frac + \frac \sum_^ \frac + \frac \sum_^ \frac \\ &= \ln\left(2\right)+\frac. \end

{{num-stub Figurate numbers