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.

2, 12, 1236, 4215840, 274888655360, 288230395747516416, 4835703278617945885900800, 1298074214633711825063417305104384, 5575186299632655786027352001453113896599552, 383123885216472214589587083913232677066440014825521152

Offset: 0

Author

Peter Kagey, Apr 21 2024

Keywords

• nonn

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.

Examples

			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 |                 |   |
        +---+                 +---+
		

Links

• Eric Weisstein's World of Mathematics, Centered Square Number.

Crossrefs

Cf. A000124, A001844, A002061, A014848.

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)).