Pyramidal Numbers

Triangular numbers are constructed starting with a single dot, then a triangle with a dot at each vertex, then a larger triangle with a dot between each two dots and so one. If these triangles are assembled into a pyramid, then the numbers of dots are called pyramidal numbers.

pyramidal numbers

Base Number of Dots
Triangle
\[\frac{1}{6}n(n+1)(n+2)\]
Square
\[\frac{1}{6}n(n+1)(2n+1)\]
Pentagon
\[\frac{1}{2}n^2(n+1)\]
Hexagon
\[\frac{1}{6}n(n+1)(4n-1)\]
Heptagon
\[\frac{1}{6}n(n+1)(5n-2)\]
Octagon
\[\frac{1}{2}n(n+1)(2n-1)\]
n - gon
\[\frac{1}{6}n(n+1)(n(r-2)-(r-5))\]

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