Figurate numbers take the shape of regular two-dimensional polygons and three-dimensional polyhedra.
Each type of figurate number has a formula associated with it. For instance, n² generates square numbers. The first few square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. . . .
The sixth square: 6² = 36
(n² +n)/2 generates triangle numbers.
The fifth triangle: (5² +5)/2 = 15
(6n² 6n +1) generates star (hexagram) numbers.
The third star, (6 × 3² 6 × 3 +1) = 37
There are also three-dimensional figurate numbers, such as truncated octahedrons, (16n³ 33n² + 24n 6).
The fourth truncated octahedron: (16 × 4³ 33 × 4² + 24 × 4 6) = 586
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