Centered pentagonal number on:
[Wikipedia]
[Google]
[Amazon]
A centered pentagonal number is a centered
A centered pentagonal 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
* polyg ...
that represents a pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered pentagonal number for ''n'' is given by the formula
:
The first few centered pentagonal numbers are
1, 6, 16, 31, 51, 76,
106, 141, 181, 226, 276
__NOTOC__
Year 276 ( CCLXXVI) was a leap 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 Tacitus and Aemilianus (or, less frequently, year 1029 ...
,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
, 391, 456, 526, 601,
681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976 .
Properties
*The parity of centered pentagonal numbers follows the pattern odd-even-even-odd, and in base 10 the units follow the pattern 1-6-6-1. *Centered pentagonal numbers follow the followingRecurrence relations
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 ...
:
:
:
*Centered pentagonal numbers can be expressed using Triangular Numbers
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 i ...
:
:
See also
*Pentagonal number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
*Polygonal number In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
Definition and examples
T ...
*Centered polygonal number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side i ...
External links
* Figurate numbers {{num-stub