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A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.

%I A273461 #41 Dec 19 2019 14:20:39
%S A273461 1,2,9,40,484,9717,338724,21624680,2504301849,520443847520,
%T A273461 195145309791364,131850659243316222,160668896658179472676,
%U A273461 352891729183598844656996,1397187513066371784602204416,9972288382286063615850619475640
%N A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.
%C A273461 In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away. This reaction creates a third "infinite" source block in the unoccupied intermediate block, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.
%C A273461 A placement of water at several positions in an n X n board is said to be *stable* if no infinite water physics can in fact occur (under otherwise optimal conditions). This means that the total quantity of water in the system is held constant.
%C A273461 In short, no two source blocks can be graph-distance 2 from each other. - _Gus Wiseman_, Nov 27 2019
%C A273461 Often incorrectly described as cellular automata, the observed behaviors of liquids within a board are inseparable in certain ways from states of affair outside of the board and events outside of the system. This aspect of Minecraft is poorly understood.
%H A273461 EthosLab, <a href="https://youtu.be/9uIRD9sMBU0">Minecraft - Tutorial: Water</a>
%H A273461 Wikipedia, <a href="https://en.wikipedia.org/wiki/Distance_(graph_theory)">Distance (graph theory)</a>
%H A273461 Gus Wiseman, <a href="/A273461/a273461.png">Example of a physically stable arrangement of water source-blocks (n=11)</a>
%H A273461 Christopher Cormier, <a href="/A273461/a273461.cs.txt">C# Program</a>
%e A273461 a(2) = 9: {{}, {(2,2)}, {(2,1)}, {(2,1),(2,2)}, {(1,2)}, {(1,2),(2,2)}, {(1,1)}, {(1,1),(2,1)}, {(1,1),(1,2)}}.
%t A273461 stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t A273461 allflows[n_]:=stableSets[Join@@Array[List,{n,n}],Function[{v,w},Plus@@Abs/@(w-v)===2]];
%t A273461 Table[Length[allflows[i]],{i,6}] (* _Gus Wiseman_, May 23 2016 *)
%Y A273461 The one-dimensional version is A006498.
%Y A273461 Dominated by A329871.
%Y A273461 Cf. A002416, A005251, A027624, A114901.
%K A273461 nonn
%O A273461 0,2
%A A273461 _Gus Wiseman_, May 23 2016
%E A273461 a(7) from _Tae Lim Kook_, May 25 2016
%E A273461 a(8) from _Tae Lim Kook_, May 29 2016
%E A273461 a(7)-a(8) corrected by _Christopher Cormier_, Dec 17 2019
%E A273461 a(9)-a(15) from _Christopher Cormier_, Dec 19 2019