A unicursal circuit is a circuit that traverses every graph edge of a graph exactly once. Examples of curves that can be traced unicursally are the Mohammed sign and unicursal hexagram.
For 
fixed points in cyclic order
on a circle, let 
count closed polygonal routes that visit each point
exactly once and do not join neighboring points on the
circle. Routes are identified under a change of starting
point or reversal of direction, while the positions on
the circle remain fixed. With the conventions 
and 
, the numbers of these unicursal polygrams for 
, 2, ... are 1, 0, 0, 0, 1, 3, 23, 177, 1553, ... (OEIS A002816). For 
, these are given by the sum
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For 
,
this sequence also satisfies the recurrence equation
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