centered hexagonal number

A centered hexagonal number, or hex number is a figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot equidistantly. The centered hexagonal number for $n$ is given by the formula $1+6\left(\frac{1}{2}n(n+1)\right)$. In other words, the centered hexagonal number for $n$ is the triangular number for $n$ multiplied by 6, then add 1.

The first few centered hexagonal numbers are: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, … listed in A003215 of Sloane’s OEIS.

To find centered hexagonal numbers besides 1 that are also triangular numbers or squares, it is necessary to solve Diophantine equations. By solving the Diophantine equation $\frac{1}{2}m(m+1)=3n^2+3n+1,$ we learn that 91, 8911 and 873181 are numbers that are both centered hexagonal numbers and triangular numbers (they grow very large after that), while solving the Diophantine equation $m^2=3n^2+3n+1,$ we learn that 169 and 32761 are centered hexagonal numbers that are also squares.

The sum of the first $n$ centered hexagonal numbers is $n^3$. The difference between ${(2n)}^2$ and the $n$th centered hexagonal number is a number of the form $n^2+3n-1$, while the difference between $(2n-1)){}^2$ and the $n$th centered hexagonal number is an oblong number.