centered hexagonal number

A centered hexagonal number, or hex number is a figurate numberMathworldPlanetmath 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(12n(n+1)). In other words, the centered hexagonal number for n is the triangular numberMathworldPlanetmath 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 equationsMathworldPlanetmath. By solving the Diophantine equation 12m(m+1)=3n2+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 m2=3n2+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 n3. The difference between (2n)2 and the nth centered hexagonal number is a number of the form n2+3n-1, while the difference between (2n-1))2 and the nth centered hexagonal number is an oblong number.

Title centered hexagonal number
Canonical name CenteredHexagonalNumber
Date of creation 2013-03-22 16:34:35
Last modified on 2013-03-22 16:34:35
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11D09
Synonym hex number