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%I #41 Nov 14 2023 11:36:33
%S 0,1,13,218,11506,2301877,1732082741,4872949974666,51016818604894742,
%T 1980555831431088025753,284374318545830329487309785,
%U 150730745416633777472365437495914,294516896499779486414143877573183893666,2119097214294718323017954923662829194285541981
%N Number of nonzero n X n binary arrays with all 1's connected.
%C Old name was: "Number of n X n checkerboards in which the set of red squares is edge connected".
%C Also the number of connected induced (non-null) subgraphs of the n X n grid graph P_n x P_n. - _Eric W. Weisstein_, May 01 2017
%H Andrew Howroyd, <a href="/A059525/b059525.txt">Table of n, a(n) for n = 0..16</a>
%H Stijn Cambie, Jan Goedgebeur, and Jorik Jooken, <a href="https://arxiv.org/abs/2311.00075">The maximum number of connected sets in regular graphs</a>, arXiv:2311.00075 [math.CO], 2023.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InducedSubgraph.html">Induced Subgraph</a>
%t Table[Count[Subgraph[GridGraph[{n, n}], #] & /@ Subsets[Range[n^2], {1, Infinity}], _?ConnectedGraphQ], {n, 4}] (* _Eric W. Weisstein_, May 01 2017 *)
%Y Main diagonal of A287151.
%Y Cf. A059021, A020873 (wheel), A059020 (ladder), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).
%K nonn,hard
%O 0,3
%A _David Radcliffe_, Jan 21 2001
%E One more term from _John W. Layman_, Jan 25 2001
%E More terms from _R. H. Hardin_, Feb 28 2002
%E Clearer name from _R. H. Hardin_, Jul 06 2009
%E a(8)-a(9) from _Giovanni Resta_, May 03 2017
%E a(10)-a(13) from _Andrew Howroyd_, May 20 2017
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