Editing Centered hexagonal number (section)

== Properties ==
[[File:visual_proof_centered_hexagonal_numbers_sum.svg|thumb|[[Proof without words]] of the sum of the first ''n'' hex numbers by arranging ''n''<sup>3</sup> semitransparent balls in a cube and viewing along a [[space diagonal]] &ndash; colour denotes cube layer and line style denotes hex number]]
In [[base 10]] one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with [[Periodic sequence|period]] 5).
This follows from the [[Triangular number#Other properties|last digit of the triangle numbers]] {{OEIS|id=A008954}} which repeat 0-1-3-1-0 when taken modulo 5.
In [[base 6]] the rightmost digit is always 1: 1<sub>6</sub>, 11<sub>6</sub>, 31<sub>6</sub>, 101<sub>6</sub>, 141<sub>6</sub>, 231<sub>6</sub>, 331<sub>6</sub>, 441<sub>6</sub>...
This follows from the fact that every centered hexagonal number modulo 6 (=10<sub>6</sub>) equals 1.

The sum of the first {{mvar|n}} centered hexagonal numbers is {{math|[[cube (algebra)|''n''<sup>3</sup>]]}}. That is, centered hexagonal [[pyramidal number]]s and [[cubic number|cubes]] are the same numbers, but they represent different shapes.  Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the [[figurate number#Gnomon|gnomon]] of the cubes. (This can be seen geometrically from the diagram.)  In particular, [[prime number|prime]] centered hexagonal numbers are [[cuban prime]]s.

The difference between {{math|(2''n'')<sup>2</sup>}} and the {{mvar|n}}th centered hexagonal number is a number of the form {{math|3''n''<sup>2</sup> + 3''n'' − 1}}, while the difference between {{math|(2''n'' − 1)<sup>2</sup>}} and the {{mvar|n}}th centered hexagonal number is a [[pronic number]].