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 is given by the formula . In other words, the centered hexagonal number for is the triangular number for 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 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 we learn that 169 and 32761 are centered hexagonal numbers that are also squares.
The sum of the first centered hexagonal numbers is . The difference between and the th centered hexagonal number is a number of the form , while the difference between and the th centered hexagonal number is an oblong number.
|Title||centered hexagonal number|
|Date of creation||2013-03-22 16:34:35|
|Last modified on||2013-03-22 16:34:35|
|Last modified by||PrimeFan (13766)|