This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
(* Version 12.0 or higher needed *)
ClearAll[ShowRoute,MakeMove,FindSequence]
knightjump=Select[Tuples[Range[-2,2],2],Norm[#]==Sqrt[5]&];
ShowRoute[output_Association]:=Module[{colors},colors=(ColorData["Rainbow"]/@Subdivide[Length[output["Coordinates"]]-1.0]);
Graphics[{Line[output["Coordinates"],VertexColors->colors],Disk[Last@output["Coordinates"],0.2],Style[Disk[Last[output["Coordinates"]]+#,0.2]&/@knightjump,Purple]}]]
MakeMove[spiral_Association,visited_List]:=Module[{poss,hj},poss=Table[Last[Last[visited]]+hj,{hj,knightjump}];
poss=DeleteMissing[{spiral[#],#}&/@poss,1,\[Infinity]];
poss=Select[poss,FreeQ[visited[[All,2]],Last[#]]&];
If[Length[poss]>0,First[TakeSmallestBy[poss,First,1]],Missing[]]]
FindSequence[start_:{0,0},grid_]:=Module[{positions,j,next},positions={{grid[start],start}};
PrintTemporary[Dynamic[j]];
Do[next=MakeMove[grid,positions];
If[next=!=Missing[],AppendTo[positions,next],Break[];],{j,\[Infinity]}];
<|"Coordinates"->positions[[All,2]],"Indices"->positions[[All,1]]|>]
grid=ResourceFunction["LatticePointsArrangement"]["DiagonalZigZagEastQ34",20000];
grid=Association[MapIndexed[#1->#2[[1]]&,grid]];
ShowRoute[fs=FindSequence[{0,0},grid]]
fs
fs["Indices"]
ListPlot[fs["Indices"]]
Between 4 and 8, the shortest route is through 12 (2*6); it takes only two steps:
.
1 2 3 4 5 6 7 8
+------+------+------+------+------+------+------+------+
| | | | | | | | |
1 | 1 | 2 | 3 | *4* | 5 | 6 | 7 | .*8* |
| | | | |. | | . | |
+------+------+------+------+---.--+------+-.----+------+
| | | | | . .| | |
2 | 2 | 4 | 6 | 8 | 10 | *12* | 14 | 16 |
| | | | | | | | |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 |
| | | | | | | | |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 |
| | | | | | | | |
+------+------+------+------+------+------+------+------+
.
Between 32 and 36, there are several routes that take only three jumps. We choose 32,18,16,36 because the sum of intermediate numbers is the least.
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