%I #14 Apr 24 2024 03:05:48

%S 2,12,1236,4215840,274888655360,288230395747516416,

%T 4835703278617945885900800,1298074214633711825063417305104384,

%U 5575186299632655786027352001453113896599552,383123885216472214589587083913232677066440014825521152

%N The number of ways of 2-coloring the diagram for the n-th centered square number up to rotation and reflection of the square.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>.

%F 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)).

%e For n=1 the a(1) = 12 inequivalent colorings are:

%e 0 1 1 0 1 1

%e 0 0 0, 0 0 0, 1 0 0, 1 0 1, 1 0 1, and 1 0 1,

%e 0 0 0 0 0 1

%e along with

%e 1 0 0 1 0 0

%e 1 1 1, 1 1 1, 0 1 1, 0 1 0, 0 1 0, and 0 1 0.

%e 1 1 1 1 1 0

%e 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.

%e +---+ +---+

%e | 1 | | 1 |

%e +---+---+---+ +---+---+---+

%e | 1 | 1 | | | 1 | | 1 |

%e +---+---+---+---+---+ +---+---+---+---+---+

%e | 1 | | | 1 | | | 1 | 1 | | | 1 |

%e +---+---+---+---+---+ +---+---+---+---+---+

%e | 1 | | | | | 1 | |

%e +---+---+---+ +---+---+---+

%e | 1 | | |

%e +---+ +---+

%Y Cf. A000124, A001844, A002061, A014848.

%K nonn

%O 0,1

%A _Peter Kagey_, Apr 21 2024