Editing Centered polygonal number

{{Short description|Class of series of figurate numbers, each having a central dot}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
[[File:visual_proof_centered_polygonal_numbers.svg|thumb|upright|[[Proof without words|Proof]] that each centered ''k''-gonal number is ''k'' times the previous triangular number, plus 1]]
In [[mathematics]], the '''centered polygonal numbers''' are a class of series of [[figurate number]]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]], plus 1. This can be formalized by the expression <math>\frac{kn(n+1)}{2} +1</math>, 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 <math>\frac{4n(n+1)}{2} +1</math>.

These series consist of the
*[[centered triangular number]]s 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... ({{OEIS2C|id=A005448}}),
*[[centered square number]]s 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... ({{OEIS2C|id=A001844}}), which are exactly the sum of consecutive squares, i.e., n^2 + (n - 1)^2.
*[[centered pentagonal number]]s 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, ... ({{OEIS2C|id=A005891}}),
*[[centered hexagonal number]]s 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... ({{OEIS2C|id=A003215}}), which are exactly the difference of consecutive cubes, i.e. ''n''<sup>3</sup> − (''n'' − 1)<sup>3</sup>,
*[[centered heptagonal number]]s 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, ... ({{OEIS2C|id=A069099}}),
*[[centered octagonal number]]s 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, ... ({{OEIS2C|id=A016754}}), which are exactly the [[Odd number|odd]] [[Square number|squares]],
*[[centered nonagonal number]]s 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, ... ({{OEIS2C|id=A060544}}), which include all even [[perfect number]]s except 6,
*[[centered decagonal number]]s 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, ... ({{OEIS2C|id=A062786}}),
*centered hendecagonal numbers 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, ... ({{OEIS2C|id=A069125}}),
*centered dodecagonal numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, ... ({{OEIS2C|id=A003154}}), which are also the [[star number]]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]].

{|
|- align="center" valign="top"
! centered<br>triangular<br>number
! centered<br>square<br>number
! centered<br>pentagonal<br>number
! centered<br>hexagonal<br>number
|-
| [[Image:Centered_triangular_number_19.svg|100px]]
| [[Image:Centered_square_number_25.svg|100px]]
| [[Image:Centered_pentagonal_number_31.svg|100px]]
| [[Image:Hex_number_37.svg|100px]]
|}

=== Centered square numbers ===

{|
|- align="center" valign="top"
! 1
! &nbsp; &nbsp;
! 5
! &nbsp; &nbsp;
! 13
! &nbsp; &nbsp;
! 25
|- align="center" valign="middle" style="line-height: 0;"
|[[Image:redDot.svg|16px]]
| &nbsp; &nbsp;
|[[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]]
| &nbsp; &nbsp;
|[[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]]
| &nbsp; &nbsp;
|[[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:redDot.svg|16px]]<br>[[Image:redDot.svg|16px]]
|}

=== Centered hexagonal numbers ===

{| style="line-height: 1em;"
!  1 !! &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
!  7 !! &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
! 19 !! &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
! 37
|- align="center" valign="middle"
|[[Image:RedDotX.svg|16px|*]]
||
|[[Image:RedDot.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]
||
|[[Image:RedDot.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]
||
|[[Image:RedDot.svg|16px|*]][[Image:RedDot.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDot.svg|16px|*]][[Image:RedDot.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]
|}
[[File:visual_proof_centered_hexagonal_numbers_sum.svg|thumb|As the sum of the first ''n'' hex numbers is ''n''<sup>3</sup>, the ''n''-th hex number is {{math|''n''<sup>3</sup> &minus; (''n''&minus;1)<sup>3</sup>}}]]

== 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''&minus;1)th triangular number around a central point; therefore, the ''n''th centered ''k''-gonal number is equal to

:<math>C_{k,n} =\frac{kn}{2}(n-1)+1.</math>

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)<sup>2</sup>.

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:

:<math>\frac{k^2}{2}(k-1)+1</math>

which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.

Whereas a [[prime number]] ''p'' cannot be a [[polygonal number]] (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]]). Since all [[centered octagonal number]]s are also [[square number]]s, and all [[centered nonagonal number]]s are also [[triangular number]]s (and not equal to 3), thus both of them cannot be prime numbers.

== Sum of reciprocals ==

The [[Summation|sum]] of [[Multiplicative inverse|reciprocal]]s for the centered ''k''-gonal numbers is<ref>[https://oeis.org/wiki/Centered_polygonal_numbers#Table_of_related_formulae_and_values centered polygonal numbers in OEIS wiki, content "Table of related formulae and values"]</ref>

:<math>\frac{2\pi}{k\sqrt{1-\frac{8}{k}}}\tan\left(\frac{\pi}{2}\sqrt{1-\frac{8}{k}}\right)</math>, if ''k'' ≠ 8

:<math>\frac{\pi^2}{8}</math>, if ''k'' = 8

== References ==

{{reflist}}

*{{cite book|author=[[Neil Sloane]] & [[Simon Plouffe]]|title=''The Encyclopedia of Integer Sequences''|publisher=San Diego: Academic Press|year=1995}}: Fig. M3826
*{{mathworld|urlname=CenteredPolygonalNumber|title=Centered polygonal number}}
*{{cite book|author=F. Tapson|title=The Oxford Mathematics Study Dictionary|publisher=Oxford University Press|year=1999|pages=88–89|edition=2nd|isbn=0-19-914-567-9}}

{{Classes of natural numbers}}

[[Category:Figurate numbers]]