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A341195 Squares visited by knight moves on a diagonally back and forth numbered board in two quadrants and moving to the lowest available unvisited square at every step. 1
1, 11, 7, 2, 6, 12, 9, 3, 5, 14, 8, 4, 18, 33, 21, 29, 24, 26, 47, 10, 23, 13, 19, 16, 38, 34, 17, 15, 20, 30, 42, 56, 45, 28, 22, 31, 41, 58, 44, 32, 40, 35, 37, 62, 66, 36, 39, 60, 68, 63, 65, 98, 102, 64, 67, 61, 70, 93, 43, 55, 46, 27, 49, 52, 25, 51, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Board is numbered as follows:

.  17 16 5  4  1  2  9  10 .  .

.  .  18 15 6  3  8  11 24 .  .

.  .  .  19 14 7  12 23 .  .  .

.  .  .  .  20 13 22 .  .  .  .

.  .  .  .  .  21 .  .  .  .  .

.  .  .  .  .  .  .  .  .  .  .

This sequence is finite: At step 4408 square 4077 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..4408

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

MATHEMATICA

(* 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"]]

CROSSREFS

Cf. A316588, A316328, A316667, A337170.

Sequence in context: A241308 A038320 A010185 * A005603 A109828 A048981

Adjacent sequences:  A341192 A341193 A341194 * A341196 A341197 A341198

KEYWORD

nonn,fini,full,changed

AUTHOR

Sander G. Huisman, Feb 06 2021

STATUS

approved

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Last modified February 27 11:38 EST 2021. Contains 341656 sequences. (Running on oeis4.)