a(1) = 8. The single walk of length 4 is:
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+---+
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This can be taken in 8 different ways on a square lattice, giving a total 1*8 = 8.
a(2) = 0 as there is no closed-loop path consisting of 8 steps.
a(3) = 24. There is one walk, ignoring reflection and rotations, with a length of 12. The walk is:
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This can be walked in 3 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 3*8 = 24.
a(4) = 64. There is one walk, with indistinct reflections and rotations, with a length of 16. The walk is:
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+---+ +---+
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+---+ +---+
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+---+ +---+
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+---+
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This can be walked in 8 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 8*8 = 64.
.
a(5) = 360. There are four walks, with indistinct reflections and rotations, with a length of 20. The walks, and the different ways they can be taken, are:
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+---+ +---+ +---+ +---+
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+---+ +---+ +---+ +---+
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+---+ +---+ +---+ +---+
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+---+ +---+ +---+ +---+
| | x 10 | | x 20
+---+ +---+
+---+ +---+
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+---+ +---+ +---+ +---+
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+---+ +---+ +---+ +---+
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+---+ +---+ +---+ +---+
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+---+ +---+ +---+ +---+
| | x 5 | | x 10
+---+ +---+
.
Each of these can be walked in 8 different ways on a square lattice, giving a total number of 8*(10+20+5+10) = 360.
See the attached text file for images of the closed-loops for n=1 to n=11.
