A centered decagonal number is a
centered 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 term can mean
* polygon ...
that represents a
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 ...
with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by the formula
:
Thus, the first few centered decagonal numbers are
:
1,
11,
31,
61,
101,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
, 211, 281, 361, 451, 551, 661, 781,
911
911 or 9/11 may refer to:
Dates
* AD 911
* 911 BC
* September 11
** 9/11, the September 11 attacks of 2001
** 11 de Septiembre, Chilean coup d'état in 1973 that outed the democratically elected Salvador Allende
* November 9
Numbers
* 91 ...
, 1051, ...
Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can be reckoned by multiplying the (''n'' − 1)th
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
by ''k'', 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in 1.
Another consequence of this relation to triangular numbers is the simple
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
for centered decagonal numbers:
:
where
:
Generating Function
The generating function of the centered decagonal number is
Continued fraction forms
has the
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
expansion
n-3;
See also
*
rdinarydecagonal number
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 number ...
References
{{Classes of natural numbers
Figurate numbers