Centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or hex number,[1][2] is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers:
Centered hexagonal numbers should not be confused with cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex.
The sequence of hexagonal numbers starts out as follows (sequence A003215 in the OEIS):
- 397, 469, 547, 631, 721, 817, 919.
Formula
The nth centered hexagonal number is given by the formula[2]
Expressing the formula as
shows that the centered hexagonal number for n is 1 more than 6 times the (n − 1)th triangular number.
In the opposite direction, the index n corresponding to the centered hexagonal number can be calculated using the formula
This can be used as a test for whether a number H is centered hexagonal: it will be if and only if the above expression is an integer.
Recurrence and generating function
The centered hexagonal numbers satisfy the recurrence relation[2]
From this we can calculate the generating function . The generating function satisfies
The latter term is the Taylor series of , so we get
and end up at
Properties
In
The sum of the first n centered hexagonal numbers is
The difference between (2n)2 and the nth centered hexagonal number is a number of the form 3n2 + 3n − 1, while the difference between (2n − 1)2 and the nth centered hexagonal number is a pronic number.
Applications
Centered hexagonal numbers have practical applications in
]Many segmented mirror reflecting telescopes have primary mirrors comprising a centered hexagonal number of segments (neglecting the central segment removed to allow passage of light) to simplify the control system.[3] Some examples:
Telescope | Number of segments |
Number missing |
Total | n-th centered hexagonal number |
---|---|---|---|---|
Giant Magellan Telescope | 7 | 0 | 7 | 2 |
James Webb Space Telescope | 18 | 1 | 19 | 3 |
Gran Telescopio Canarias | 36 | 1 | 37 | 4 |
Guido Horn d'Arturo's prototype | 61 | 0 | 61 | 5 |
Southern African Large Telescope | 91 | 0 | 91 | 6 |
References
- ^ Hindin, H. J. (1983). "Stars, hexes, triangular numbers and Pythagorean triples". J. Rec. Math. 16: 191–193.
- ^ ISBN 978-981-4355-48-3.
- ^ Mast, T S, and Nelson, J E. Figure control for a segmented telescope mirror. United States: N. p., 1979. Web. doi:10.2172/6194407.