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

1, 2, 9, 40, 484, 9717, 338724, 21624680, 2504301849, 520443847520, 195145309791364, 131850659243316222, 160668896658179472676, 352891729183598844656996, 1397187513066371784602204416, 9972288382286063615850619475640

Offset: 0

Author

Gus Wiseman, May 23 2016

Keywords

• nonn

Comments

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.

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.

In short, no two source blocks can be graph-distance 2 from each other. - Gus Wiseman, Nov 27 2019

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.

Examples

			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)}}.
		

Links

• EthosLab, Minecraft - Tutorial: Water
• Wikipedia, Distance (graph theory)
• Gus Wiseman, Example of a physically stable arrangement of water source-blocks (n=11)
• Christopher Cormier, C# Program

Crossrefs

The one-dimensional version is A006498.

Dominated by A329871.

Cf. A002416, A005251, A027624, A114901.

Programs

Mathematica

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]]]];
allflows[n_]:=stableSets[Join@@Array[List,{n,n}],Function[{v,w},Plus@@Abs/@(w-v)===2]];
Table[Length[allflows[i]],{i,6}] (* Gus Wiseman, May 23 2016 *)

Extensions

a(7) from Tae Lim Kook, May 25 2016

a(8) from Tae Lim Kook, May 29 2016

a(7)-a(8) corrected by Christopher Cormier, Dec 17 2019

a(9)-a(15) from Christopher Cormier, Dec 19 2019