A plane path on a set of equally spaced lattice points, starting at the origin, where the first step is one unit to the north or south, the second step is two units to the east or west, the third is three units to the north or south, etc., and continuing until the origin is again reached. No crossing or backtracking is allowed. The simplest golygon is (0, 0), (0, 1), (2, 1), (2, ), (, ), (, ), (, ), (, 0), (0, 0).
A golygon can be formed if there exists an even integer such that
(1)
| |||
(2)
|
(Vardi 1991). Gardner proved that all golygons are of the form . The number of golygons of length (even), with each initial direction counted separately, is the product of the coefficient of in
(3)
|
with the coefficient of in
(4)
|
The number of golygons of length for the first few is therefore 4, 112, 8432, 909288, ... (OEIS A006718), and is asymptotic to
(5)
|
(Sallows et al. 1991, Vardi 1991).