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A337170 Squares visited by knight moves on a diagonally back and forth numbered board and moving to the lowest available unvisited square at every step. 2
1, 8, 4, 2, 13, 3, 6, 9, 11, 19, 5, 7, 26, 16, 14, 31, 15, 18, 12, 21, 24, 10, 23, 33, 39, 20, 22, 34, 25, 17, 28, 47, 43, 29, 27, 32, 40, 35, 37, 53, 57, 36, 58, 52, 38, 55, 80, 76, 56, 54, 59, 51, 42, 30, 45, 48, 62, 70, 44, 46, 64, 49, 41, 72, 60, 50, 63 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Board is numbered as follows:

1  3  4  10 11 .

2  5  9  12 .  .

6  8  13 19 .  .

7  14 18 .  .  .

15 17 .  .  .  .

16 .  .  .  .  .

This sequence is finite: At step 343 square 276 is visited, after which there are no unvisited squares within one knight move.

LINKS

Sander G. Huisman, Table of n, a(n) for n = 1..343

N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (January, 2019)

MATHEMATICA

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]}]]

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"]["DiagonalZigZagEastQ4", 10000];

grid=Association[MapIndexed[#1->#2[[1]]&, grid]];

ShowRoute[fs=FindSequence[{0, 0}, grid]]

fs

fs["Indices"]

ListPlot[fs["Indices"]]

CROSSREFS

Cf. A316588, A316328, A316667.

Sequence in context: A099286 A089729 A173670 * A050135 A109595 A256782

Adjacent sequences:  A337167 A337168 A337169 * A337171 A337172 A337173

KEYWORD

nonn,fini,full,look

AUTHOR

Sander G. Huisman, Jan 28 2021

STATUS

approved

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Last modified April 13 19:10 EDT 2021. Contains 342939 sequences. (Running on oeis4.)