By Geeks out of the Box Posted on

The Formula for Pentagonal Numbers

Here at the bottom of the whiteboard: deriving the formula for calculating pentagonal numbers: n(n+1)/2 + n(n-1). The formula for triangular numbers is the same as the formula for calculating the sum of consecutive integers n(n+1)/2 because the number of distinct dots in every row constructing a triangle corresponds to consecutive integers. And the formula for square numbers is the same as the formula for calculating the sum of consecutive odd numbers, n^2.

A pentagonal number extends the concept of triangular and square numbers, but the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in the pentagon with sides of n dots, when the pentagons are overlaid sharing one vertex.

This is the third pentagonal number, 12. Simon: “It helps to think of this as a pentagon with a side n”.
This is the fourth pentagonal number, 22. Simon types: Lay the two mikado sticks at an angle of 108 degrees. Place a checker in the angle. Try to enclose the checker with as few checkers as possible. You’ll find you need 4 more. Now, try to enclose this figure with as few checkers as possible. You’ll need 7. Now, try to enclose this figure, again, with as few checkers as possible. You’ll need 10 this time. And so on. At each stage, the total number of checkers are called the pentagonal numbers.
Simon teaching pentagonal numbers and the formula to calculate them to Neva, using a set of Go and mikado sticks.
This is the formula for centered hexagonal numbers. These arise if you place the sticks at a 120 degree angle, or just throw away the sticks entirely.

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