# Star number

A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on. The nth star number 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 (sequence A003154 in OEIS).

Geometrically, the nth star number is made up of a central point and 12 copies of the (n-1)th triangular number — making it numerically equal to the nth centred dodecagonal number, but differently arranged.

The digital root of a star number is always 1 or 4. The last two digits of a star number in base 10 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.

Not many star numbers are also square. 1 and 121 are the only two such numbers in the list given above. Square star numbers can be searched for with the Diophantine equation 2x2 + 1 = 3y2.

All star numbers can be constructed by multiplying a triangular number by 12 and adding one. Hence, all star numbers are odd.

Confusingly, the term "star number" or "stellate number" is occasionally used to refer to octagonal numbers.

A star prime is a star number that is prime. The first few star primes (sequence A083577 in OEIS) are

13, 37, 73, 181, 337, 433, 541, 661, 937. 