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 ancient Greek mathemat ...
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°. Regular decagon
A '' regular decagon'' has a ...
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
101 may refer to:
*101 (number), the number
* AD 101, a year in the 2nd century AD
* 101 BC, a year in the 2nd century BC
It may also refer to:
Entertainment
* ''101'' (album), a live album and documentary by Depeche Mode
* "101" (song), a 19 ...
,
151, 211, 281, 361, 451, 551, 661, 781,
911
911, 9/11 or Nine Eleven may refer to:
Dates
* AD 911
* 911 BC
* September 11
** The 2001 September 11 attacks on the United States by al-Qaeda, commonly referred to as 9/11
** 11 de Septiembre, Chilean coup d'état in 1973 that ousted the ...
, 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
:
Relation to other sequences
* N is a Centered decagonal number iff 20N + 5 is a
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 ...
.
Generating Function
The generating function of the centered decagonal number is
Continued fraction forms
has the
simple continued fraction
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
n-3;
See also
*
rdinary decagonal number
References
{{Classes of natural numbers
Figurate numbers