Hex Number

The Centered Hexagonal Number given by

$$H_n=1+6T_n=2H_{n-1}-H_{n-2}+6=3n^2-3n+1,$$

where $T_n$ is the $n$th Triangular Number. The first few hex numbers are 1, 7, 19, 37, 61, 91, 127, 169, ... (Sloane's A003215). The Generating Function of the hex numbers is

$${x(x^2+4x+1)\over(1-x)^3}=x+7x^2+19x^3+37x^4+\ldots.$$

The first Triangular hex numbers are 1 and 91, and the first few Square ones are 1, 169, 32761, 6355441, ... (Sloane's A006051). Square hex numbers are obtained by solving the Diophantine Equation

$$3x^2+1=y^2.$$

The only hex number which is Square and Triangular is 1. There are no Cubic hex numbers.

See also Magic Hexagon, Centered Square Number, Star Number, Talisman Hexagon

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 41, 1996.

Gardner, M. ``Hexes and Stars.'' Ch. 2 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, 1988.

Hindin, H. ``Stars, Hexes, Triangular Numbers, and Pythagorean Triples.'' J. Recr. Math. 16, 191-193, 1983-1984.

Sloane, N. J. A. Sequences A003215/M4362 and A006051/M5409 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.