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(12n(n+1)). 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 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 |
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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 |