A golygon is a closed plane path on equally spaced lattice points, starting at the origin, with successive steps
of lengths 1, 2, 3, ... and a right-angle turn after each step, including the last.
The path may cross itself, touch itself at corners, or overlap along segments (Sallows
et al. 1991, p. 315). The simplest golygon is (0, 0), (0, 1), (2, 1),
(2, 
), (
, 
),
(
, 
), (
,
), (
, 0), (0, 0).
A golygon with 
sides can be formed iff 
is an even positive
integer for which signs can be chosen so that
 |  |  |
(1)
|
 |  |  |
(2)
|
(Vardi 1991). Gardner proved that the number of sides must be of the form 
,
and golygons exist for every positive integer 
(Sallows et al. 1991, pp. 317-318).
For 
a positive multiple of 8, the number
of golygons with 
sides, with the first step north or south and all sign choices counted separately,
is the product of the coefficient
of 
in
 |
(3)
|
with the coefficient of 
in
 |
(4)
|
The number 
of golygons with 
sides, for 
,
2, ..., is therefore 4, 112, 8432, 909288, ... (OEIS A006718),
and is asymptotic to
 |
(5)
|
(Sallows et al. 1991, p. 321; Vardi 1991).
