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Number of physically stable n X n placements of water source-blocks in Minecraft.
4

%I #41 Dec 19 2019 14:20:39

%S 1,2,9,40,484,9717,338724,21624680,2504301849,520443847520,

%T 195145309791364,131850659243316222,160668896658179472676,

%U 352891729183598844656996,1397187513066371784602204416,9972288382286063615850619475640

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

%C 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 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 In short, no two source blocks can be graph-distance 2 from each other. - _Gus Wiseman_, Nov 27 2019

%C 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 EthosLab, <a href="https://youtu.be/9uIRD9sMBU0">Minecraft - Tutorial: Water</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Distance_(graph_theory)">Distance (graph theory)</a>

%H Gus Wiseman, <a href="/A273461/a273461.png">Example of a physically stable arrangement of water source-blocks (n=11)</a>

%H Christopher Cormier, <a href="/A273461/a273461.cs.txt">C# Program</a>

%e 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 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 allflows[n_]:=stableSets[Join@@Array[List,{n,n}],Function[{v,w},Plus@@Abs/@(w-v)===2]];

%t Table[Length[allflows[i]],{i,6}] (* _Gus Wiseman_, May 23 2016 *)

%Y The one-dimensional version is A006498.

%Y Dominated by A329871.

%Y Cf. A002416, A005251, A027624, A114901.

%K nonn

%O 0,2

%A _Gus Wiseman_, May 23 2016

%E a(7) from _Tae Lim Kook_, May 25 2016

%E a(8) from _Tae Lim Kook_, May 29 2016

%E a(7)-a(8) corrected by _Christopher Cormier_, Dec 17 2019

%E a(9)-a(15) from _Christopher Cormier_, Dec 19 2019