Star Number

The number of cells in a generalized Chinese checkers board (or ``centered'' Hexagram).

$\begin{displaymath} S_n=6n(n+1)+1=S_{n-1}+12(n-1). \end{displaymath}$

(1)

The first few are 1, 13, 37, 73, 121, ... (Sloane's A003154). Every star number has Digital Root 1 or 4, and the final digits must be one of: 01, 21, 41, 61, 81, 13, 33, 53, 73, 93, or 37.

The first Triangular star numbers are 1, 253, 49141, 9533161, ... (Sloane's A006060), and can be computed using

$\displaystyle {\it TS}_n$	$\textstyle =$	$\displaystyle {3[(7+4\sqrt{3}\,)^{2n-1}+(7-4\sqrt{3}\,)^{2n-1}]-10\over 32}$	(2)
	$\textstyle =$	$\displaystyle 194{\it TS}_{n-1}+60-{\it TS}_{n-2}.$	(3)

The first few Square star numbers are 1, 121, 11881, 1164241, 114083761, ... (Sloane's A006061). Square star numbers are obtained by solving the Diophantine Equation

$\begin{displaymath} 2x^2+1=3y^2 \end{displaymath}$

(4)

and can be computed using

$\begin{displaymath} {\it SS}_n={[(5+2\sqrt{6}\,)^n(\sqrt{6}-2)-(5-2\sqrt{6}\,)^n(\sqrt{6}+2)]^2\over 4}. \end{displaymath}$

(5)

See also Hex Number, Square Number, Triangular Number

References

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 A003154/M4893, A006060/M5425, and A006061/M5385 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.