A decagonal number is a figurate number that extends the concept of
triangular
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- collinea ...
and
square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The usu ...
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°.
A self-intersecting ''regular decagon'' i ...
(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 number of 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
:
The first few decagonal numbers are:
:
0,
1,
10,
27,
52,
85,
126,
175
Year 175 ( CLXXV) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Piso and Iulianus (or, less frequently, year 928 '' Ab urbe condi ...
, 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
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
:
Properties
* Decagonal numbers consistently alternate
parity.
*
is the sum of the first
natural numbers congruent to 1 mod 8.
*
is number of divisors of
.
* The only decagonal numbers that are square numbers are 0 and 1.
* The decagonal numbers follow the following recurrence relations:
:
:
:
{{num-stub
Figurate numbers