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A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

%I A068416 #35 Dec 11 2024 17:11:56
%S A068416 0,6,53,627,16213,1123743,221984391,127561384993,215767063451331,
%T A068416 1082828220389781579,16209089366362071416785,
%U A068416 726438398002211876667379681,97741115155002465272674416929195,39565596445488219947994403962984729307
%N A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.
%C A068416 One of the partitions may completely surround the other. (Cf. A271802) - _Andrew Howroyd_, Apr 14 2016
%C A068416 Number of minimal edge cuts in the n X n grid graph. - _Andrew Howroyd_, Dec 11 2024
%H A068416 Anthony J. Guttmann and Iwan Jensen, <a href="/A068416/b068416.txt">Table of n, a(n) for n = 1..26</a>
%H A068416 Benjamin Fifield, Kosuke Imai, Jun Kawahara, and Christopher T. Kenny, <a href="https://imai.fas.harvard.edu/research/files/enumerate.pdf">The Essential Role of Empirical Validation in Legislative Redistricting Simulation</a>, Tech. rep., Department of Government and Department of Statistics, Harvard University (2019).
%H A068416 Anthony J. Guttmann and Iwan Jensen, <a href="https://arxiv.org/abs/2211.14482">The gerrymander sequence, or A348456</a>, arXiv:2211.14482 [math.CO], 2022.
%H A068416 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%H A068416 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalEdgeCut.html">Minimal Edge Cut</a>.
%F A068416 a(n) = A271802(n) + A140517(n-2). - _Andrew Howroyd_, Apr 14 2016
%F A068416 a(n) = A166755(n)/2. - _Andrew Howroyd_, Dec 11 2024
%e A068416 Illustration of a(2)=6:
%e A068416    11   12   12   12   11   11
%e A068416    22   12   22   11   12   21
%e A068416 Illustration of a few solutions of a(3):
%e A068416    111   112   122   111   111
%e A068416    121   111   112   212   111
%e A068416    111   111   222   222   222
%Y A068416 Cf. A068392, A166755, A271802, A068381, A068417, A113900, A348456, A358289.
%K A068416 nonn
%O A068416 1,2
%A A068416 _R. H. Hardin_, Mar 02 2002
%E A068416 a(7)-a(14) from _Andrew Howroyd_, Apr 14 2016