Inmathematics, thecentered polygonal numbers are a class of series offigurate 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 in the previous layer; so starting from the second polygonal layer, each layer of a centeredk-gonal number containsk more dots than the previous layer.

Each centeredk-gonal number in the series isk times the previoustriangular number, plus 1. This can be formalized by the expression${\displaystyle \frac{kn(n+1)}{2}+1}$, wheren 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${\displaystyle \frac{4n(n+1)}{2}+1}$.

These series consist of the

• centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (OEIS: A005448),
• centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... (OEIS: A001844), which are exactly the sum of consecutive squares, i.e.,n² + (n − 1)².
• centered pentagonal numbers 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, ... (OEIS: A005891),
• centered hexagonal numbers 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... (OEIS: A003215), which are exactly the difference of consecutive cubes, i.e.n³ − (n − 1)³,
• centered heptagonal numbers 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, ... (OEIS: A069099),
• centered octagonal numbers 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, ... (OEIS: A016754), which are exactly theoddsquares,
• centered nonagonal numbers 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, ... (OEIS: A060544), which include all evenperfect numbers except 6,
• centered decagonal numbers 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, ... (OEIS: A062786),
• centered hendecagonal numbers 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, ... (OEIS: A069125),
• centered dodecagonal numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, ... (OEIS: A003154), which are also thestar numbers,

and so on.

The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams inPolygonal number.

centered triangular number	centered square number	centered pentagonal number	centered hexagonal number

1		5		13		25

1		7		19		37

As can be seen in the above diagrams, thenth centeredk-gonal number can be obtained by placingk copies of the (n−1)th triangular number around a central point; therefore, thenth centeredk-gonal number is equal to

${\displaystyle C_{k,n}=\frac{kn}{2}(n-1)+1.}$

The difference of then-th and the (n+1)-th consecutive centeredk-gonal numbers isk(2n+1).

Then-th centeredk-gonal number is equal to then-th regulark-gonal number plus (n−1)².

Just as is the case with regular polygonal numbers, the first centeredk-gonal number is 1. Thus, for anyk, 1 is bothk-gonal and centeredk-gonal. The next number to be bothk-gonal and centeredk-gonal can be found using the formula:

${\displaystyle \frac{k^2}{2}(k-1)+1}$

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

Whereas aprime numberp cannot be apolygonal number (except the trivial case, i.e. eachp is the secondp-gonal number), many centered polygonal numbers are primes. In fact, ifk ≥ 3,k ≠ 8,k ≠ 9, then there are infinitely many centeredk-gonal numbers which are primes (assuming theBunyakovsky conjecture). Since allcentered octagonal numbers are alsosquare numbers, and allcentered nonagonal numbers are alsotriangular numbers (and not equal to 3), thus both of them cannot be prime numbers.

Thesum ofreciprocals for the centeredk-gonal numbers is^[1]

${\displaystyle \frac{2\pi }{k\sqrt{1-\frac{8}{k}}}\tan \left(\frac{\pi }{2}\sqrt{1-\frac{8}{k}}\right)}$, ifk ≠ 8

${\displaystyle \frac{{\pi }^2}{8}}$, ifk = 8

• Neil Sloane &Simon Plouffe (1995).The Encyclopedia of Integer Sequences. San Diego: Academic Press.{{cite book}}: CS1 maint: publisher location (link): Fig. M3826
• Weisstein, Eric W."Centered polygonal number".MathWorld.
• F. Tapson (1999).The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89.ISBN 0-19-914-567-9.

• Figurate numbers

• Articles with short description
• Short description is different from Wikidata
• Use American English from March 2021
• All Wikipedia articles written in American English
• Use mdy dates from March 2021
• CS1 maint: publisher location