Star Number

A star number is the number of cells in a generalized Chinese checkers board (or "centered" hexagram). Here the star numbers are indexed by positive integers , with , and are defined by

(1)

The first few for , 2, ... are 1, 13, 37, 73, 121, ... (OEIS A003154), and the generating function for the star numbers, as a formal power series or for , is

(2)

The star numbers satisfy the linear recurrence equation, with and ,

(3)

In base 10, every star number has digital root 1 or 4, and the final digit must be one of 1, 3, or 7, and the final two digits must be one of 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93.

The first few triangular star numbers are 1, 253, 49141, 9533161, ... (OEIS A006060), and can be computed using the closed form for and the recurrence relation for , with and ,

			(4)
			(5)

The indices of the corresponding triangular numbers are 1, 22, 313, 4366, 60817, ... (OEIS A028231), and of the star numbers are 1, 7, 91, 1261, 17557, ... (OEIS A156712).

The first few square star numbers are 1, 121, 11881, 1164241, 114083761, ... (OEIS A006061). The indices of the corresponding square numbers are 1, 11, 109, 1079, 10681, 105731, 1046629, ... (OEIS A054320), and of the star numbers are 1, 5, 45, 441, 4361, 43165, 427285, ... (OEIS A054318). Square star numbers are obtained by solving the Diophantine equation

(6)

for positive integers and , where is the square root and is the star index, and can be computed for using

(7)