cp's OEIS Frontend

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.

%I A372198 #14 Apr 24 2024 03:05:48
%S A372198 2,12,1236,4215840,274888655360,288230395747516416,
%T A372198 4835703278617945885900800,1298074214633711825063417305104384,
%U A372198 5575186299632655786027352001453113896599552,383123885216472214589587083913232677066440014825521152
%N 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.
%C A372198 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 A372198 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>.
%F A372198 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 A372198 For n=1 the a(1) = 12 inequivalent colorings are:
%e A372198   0      1      1      0      1          1
%e A372198 0 0 0, 0 0 0, 1 0 0, 1 0 1, 1 0 1, and 1 0 1,
%e A372198   0      0      0      0      0          1
%e A372198 along with
%e A372198   1      0      0      1      0          0
%e A372198 1 1 1, 1 1 1, 0 1 1, 0 1 0, 0 1 0, and 0 1 0.
%e A372198   1      1      1      1      1          0
%e A372198 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 A372198         +---+                 +---+
%e A372198         | 1 |                 | 1 |
%e A372198     +---+---+---+         +---+---+---+
%e A372198     | 1 | 1 |   |         | 1 |   | 1 |
%e A372198 +---+---+---+---+---+ +---+---+---+---+---+
%e A372198 | 1 |   |   | 1 |   | | 1 | 1 |   |   | 1 |
%e A372198 +---+---+---+---+---+ +---+---+---+---+---+
%e A372198     | 1 |   |   |         |   | 1 |   |
%e A372198     +---+---+---+         +---+---+---+
%e A372198         | 1 |                 |   |
%e A372198         +---+                 +---+
%Y A372198 Cf. A000124, A001844, A002061, A014848.
%K A372198 nonn
%O A372198 0,1
%A A372198 _Peter Kagey_, Apr 21 2024