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Centered polygonal numbers are a series of numbers in which layers of polygons can be drawn around a centered point. '

Example of centered polygonal numbers includes:

• A centered triangular number consisting of a central dot with three dots around it, and then additional dots in the gaps between adjacent sides. The result is a 3-sided polygon of increasing size.

  The first few centered triangular numbers (excluding the center dot) are:

  4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, ...

  To get a number in this sequence, use the formula 𝒑𝒏 = (𝟑𝒏^𝟐 + 𝟑𝒏 + 𝟐) / 𝟐.

• A centered pentagonal number consisting of a central dot with five-dots around it, and then additional dots in the gaps between adjacent sides. The result is a 5-sided polygon of increasing size.

  The first few centered pentagonal numbers (excluding the center dot) are:

  6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, ...

  To get a number in this sequence, use the formula 𝒑𝒏 = (𝟓𝒏^𝟐 + 𝟓𝒏 + 𝟐) / 𝟐.

• A centered heptagonal number consisting of a central dot with seven dots around it, and then additional dots in the gaps between adjacent sides. The result is a 7-sided polygon of increasing size.

  The first few centered heptagonal numbers (excluding the center dot) are:

  8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, ....

  To get a number in this sequence, use the formula 𝒑𝒏= (𝟕𝒏^𝟐 + 𝟕𝒏 + 𝟐) / 𝟐

• A centered hendecagonal number consisting of a central dot with eleven-dots around it, and then additional dots in the gaps between adjacent sides. The result is an 11-sided polygon of increasing size.

  The first few centered hendecagonal numbers (excluding the center dot) are:

  12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, ...

  To get a number in this sequence, use the formula 𝒑𝒏= 𝟏𝟏𝒏^𝟐 + 𝟏𝟏𝒏 + 𝟐) / 𝟐