Art with Hexagons

Other examples of centered hexagonal numbers are:

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817

They follow the formula:

$$H_n = 3n(n-1)+1$$

where $n$ is the number of crayons along one side including both corner crayons.

For this artwork, $n=17$:

$$H_{17}=3(17)(16)+1=817$$

Another way to see the pattern is as rings around a central crayon:

$$1,\quad 1+6,\quad 1+6+12,\quad 1+6+12+18,\ldots$$

Each new ring adds 6 more crayons than the previous ring.

This artwork has:

• 1 crayon at the centre
• 16 complete rings around it
• 96 crayons in the outermost ring, since $6 \times 16 = 96$

So the total number of crayons is:

$$\boxed{817}$$