The centered polygonal numbers are a class of series of
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 ...
s, 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 in the previous layer; so starting from the second polygonal layer, each layer of a centered ''k''-gonal number contains ''k'' more dots than the previous layer.
Examples
Each centered ''k''-gonal number in the series is ''k'' times the previous
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 ...
, plus 1. This can be formalized by the expression
, where ''n'' is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression
.
These series consist of the
*
centered triangular number
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.
The following ...
s 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (),
*
centered square number
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each cen ...
s 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... (),
*
centered pentagonal number
A centered pentagonal number is a centered figurate number that represents a pentagon 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 ...
s 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, ... (),
*
centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following ...
s 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... (), which are exactly the difference of consecutive cubes, i.e. ''n''
3 − (''n'' − 1)
3,
*
centered heptagonal number
A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for ''n'' is given by ...
s 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, ... (),
*
centered octagonal number
A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the od ...
s 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, ... (), which are exactly the
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
,
*
centered nonagonal number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal n ...
s 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, ... (), which include all even
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
s except 6,
*
centered decagonal number
A centered decagonal number is a centered figurate number that represents a decagon 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 ...
s 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, ... (),
*
centered hendecagonal 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 ...
s 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, ... (),
*
centered dodecagonal 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 ...
s 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, ... (), which are also the
star number
A star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on.
The ''n''th star number is given by the formula ''Sn'' = 6''n''(''n'' − 1) + 1. The ...
s,
and so on.
The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams in
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 square numbers
Centered hexagonal numbers
Formulas
As can be seen in the above diagrams, the ''n''th centered ''k''-gonal number can be obtained by placing ''k'' copies of the (''n''−1)th triangular number around a central point; therefore, the ''n''th centered ''k''-gonal number can be mathematically represented by
:
The difference of the ''n''-th and the (''n''+1)-th consecutive centered ''k''-gonal numbers is ''k''(2''n''+1).
The ''n''-th centered ''k''-gonal number is equal to the ''n''-th regular ''k''-gonal number plus (''n''-1)
2.
Just as is the case with regular polygonal numbers, the first centered ''k''-gonal number is 1. Thus, for any ''k'', 1 is both ''k''-gonal and centered ''k''-gonal. The next number to be both ''k''-gonal and centered ''k''-gonal can be found using the formula:
:
which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.
Whereas a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' cannot be a
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 ...
(except the trivial case, i.e. each ''p'' is the second ''p''-gonal number), many centered polygonal numbers are primes. In fact, if ''k'' ≥ 3, ''k'' ≠ 8, ''k'' ≠ 9, then there are infinitely many centered ''k''-gonal numbers which are primes (assuming the
Bunyakovsky conjecture
The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial f(x) in one variable with integer coefficients to give infinitely many prime values in the sequencef(1), f(2), f(3),\ldots. It was stated in 1857 by the Ru ...
). (Since all
centered octagonal number
A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the od ...
s are also
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, and all
centered nonagonal number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal n ...
s are also
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 ...
s (and not equal to 3), thus both of them cannot be prime numbers.)
Sum of reciprocals
The
sum of
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
s for the centered ''k''-gonal numbers is
centered polygonal numbers in OEIS wiki, content "Table of related formulae and values"
/ref>
:, if ''k'' ≠ 8
:, if ''k'' = 8
References
*: Fig. M3826
*
*
{{Classes of natural numbers
Figurate numbers