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URL: https://oeis.org/A372198

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A372198
The number of ways of 2-coloring the diagram for the n-th centered square number up to rotation and reflection of the square.
0
2, 12, 1236, 4215840, 274888655360, 288230395747516416, 4835703278617945885900800, 1298074214633711825063417305104384, 5575186299632655786027352001453113896599552, 383123885216472214589587083913232677066440014825521152
OFFSET
0,1
COMMENTS
The centered square numbers are given by A001844. The diagram for the n-th centered square number is the set of integer pairs (x,y) such that |x| + |y| <= n.
LINKS
Eric Weisstein's World of Mathematics, Centered Square Number.
FORMULA
a(n) = 1/8*(2^A001844(n) + 2*2^((n+1)^2) + 2*2^A000124(n) + 2^A002061(n+1) + 2*2^A014848(n+1)).
EXAMPLE
For n=1 the a(1) = 12 inequivalent colorings are:
0 1 1 0 1 1
0 0 0, 0 0 0, 1 0 0, 1 0 1, 1 0 1, and 1 0 1,
0 0 0 0 0 1
along with
1 0 0 1 0 0
1 1 1, 1 1 1, 0 1 1, 0 1 0, 0 1 0, and 0 1 0.
1 1 1 1 1 0
For n = 2, the corresponding diagram has A001844(2) = 13 boxes and a(2) = 1236 inequivalent colorings. For instance, the following two colorings are considered equivalent, because one can be transformed to the other by a reflection.
+---+ +---+
| 1 | | 1 |
+---+---+---+ +---+---+---+
| 1 | 1 | | | 1 | | 1 |
+---+---+---+---+---+ +---+---+---+---+---+
| 1 | | | 1 | | | 1 | 1 | | | 1 |
+---+---+---+---+---+ +---+---+---+---+---+
| 1 | | | | | 1 | |
+---+---+---+ +---+---+---+
| 1 | | |
+---+ +---+
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Kagey, Apr 21 2024
STATUS
approved