Polygonal Numbers

Polygonal numbers are a type of figurate numbers. They generalize triangular numbers, square numbers, etc. to any -gon.

Return the first 10 triangular numbers.

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Table[PolygonalNumber[n], {n, 1, 10}]

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Return the tenth -gonal number of several regular polygons.

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Table[PolygonalNumber[r, 10], {r, 3, 10}]

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A polygonal number counts the number of equally spaced points in a regular geometrical arrangement of this type.

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points[0, r_] := {{0, 0}}; points[n_, r_] := Flatten[Subdivide[#1, #2, n] & @@@ Partition[CirclePoints[-Last[CirclePoints[n, r]], n, r], 2, 1, 1], 1]; points[ns_List, r_] := points[#, r] & /@ ns;

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polygonalNumberGraphics[r_, n_] := With[{ps = points[Range[0, n - 1], r]}, Graphics[{PointSize[Large], Point /@ ps, Line /@ ps, Opacity[0.2], RandomColor[], Polygon /@ ps}]]

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PolygonalNumber[3, 4] polygonalNumberGraphics[3, 4]

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PolygonalNumber[3, 6] polygonalNumberGraphics[3, 6]

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PolygonalNumber[6, 5] polygonalNumberGraphics[6, 5]

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