Centered hexagonal number

In the opposite direction, the index n corresponding to the centered hexagonal number

can be calculated using the formula This can be used as a test for whether a number H is centered hexagonal: it will be if and only if the above expression is an integer.

The generating function satisfies The latter term is the Taylor series of

, so we get and end up at In base 10 one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with period 5).

This follows from the last digit of the triangle numbers (sequence A008954 in the OEIS) which repeat 0-1-3-1-0 when taken modulo 5.

This follows from the fact that every centered hexagonal number modulo 6 (=106) equals 1.

Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes.

Centered hexagonal numbers appearing in the Catan board game:
19 land tiles,
37 total tiles
Dissection of hexagonal number into six triangles with a remainder of one. The triangles can be re-assembled pairwise to give three parallelograms of n ( n −1) dots each.
Proof without words of the sum of the first n hex numbers by arranging n 3 semitransparent balls in a cube and viewing along a space diagonal – colour denotes cube layer and line style denotes hex number
Ignoring central holes, the number of mirror segments in several segmented mirror telescopes are centered hexagonal numbers