A star number, or centered 12-gonal number, is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers are played on. The star numbers for n is given by the formula 6n(n - 1) + 1. The first few star numbers are
1, 13, 37, 73, 121, 181 , 253 , 337 , 433 , 541 , 661 , 793 , 937 , 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837
The digital root of a star number is always 1 or 4. The last two digits of a star number are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93.
Not many star numbers are also triangular numbers. 1 and 253 are the only two such numbers in the list given above.
Also, not many star numbers are also square, with 1 and 121 being the only two such numbers in the list given above. Square star numbers can be searched for with the Diophantine equation
2x2 + 1 = 3y2
See also: centered hexagonal number
