Somewhere between counting labeled stamp foldings and unlabeled stamp foldings, consider folding strips of stamps with alternating colors.
Two stamps, two distinct foldings:
For labeled stamps, there are six foldings:
But for a strip of three stamps with alternating corners, half of the foldings are mirror images of another one, which we consider duplicates.
That leaves us with three stamps, three distinct foldings:
For even numbers of stamps, the counts are the same as for labeled stamps. Four stamps, 16 foldings:
For odd numbers of stamps, half of them are mirror images. Five stamps, 25 distinct foldings:
Six stamps, 144 foldings, same as labeled stamps:
Seven stamps, 231 foldings after removing mirror images:
There doesn’t seem to be an OEIS sequence for these counts (2, 3, 16, 25, 144, 231, 1392, 2268, 14060, …).
If you kind of liked this, you’ll kind of love map foldings, labeled stamp foldings, unlabeled stamp foldings, symmetric stamp foldings, or symmetric unlabeled stamp foldings.
See Martin Gardner, Wheels, Life and Other Mathematical Amusements, pp. 60–61, 1983.
Figures created with Wolfram Mathematica versions 11 and 15.
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