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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

%I

%S 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,

%T 22,34,25,17,28,47,43,29,27,32,40,35,37,53,57,36,58,52,38,55,80,76,56,

%U 54,59,51,42,30,45,48,62,70,44,46,64,49,41,72,60,50,63

%N Squares visited by knight moves on a diagonally back and forth numbered board and moving to the lowest available unvisited square at every step.

%C Board is numbered as follows:

%C 1 3 4 10 11 .

%C 2 5 9 12 . .

%C 6 8 13 19 . .

%C 7 14 18 . . .

%C 15 17 . . . .

%C 16 . . . . .

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

%H Sander G. Huisman, <a href="/A337170/b337170.txt">Table of n, a(n) for n = 1..343</a>

%H N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).

%t ClearAll[ShowRoute,MakeMove,FindSequence]

%t knightjump=Select[Tuples[Range[-2,2],2],Norm[#]==Sqrt[5]&];

%t ShowRoute[output_Association]:=Module[{colors},colors=(ColorData["Rainbow"]/@Subdivide[Length[output["Coordinates"]]-1.0]);

%t Graphics[{Line[output["Coordinates"],VertexColors->colors],Disk[Last@output["Coordinates"],0.2]}]]

%t MakeMove[spiral_Association,visited_List]:=Module[{poss,hj},poss=Table[Last[Last[visited]]+hj,{hj,knightjump}];

%t poss=DeleteMissing[{spiral[#],#}&/@poss,1,\[Infinity]];

%t poss=Select[poss,FreeQ[visited[[All,2]],Last[#]]&];

%t If[Length[poss]>0,First[TakeSmallestBy[poss,First,1]],Missing[]]]

%t FindSequence[start_:{0,0},grid_]:=Module[{positions,j,next},positions={{grid[start],start}};

%t PrintTemporary[Dynamic[j]];

%t Do[next=MakeMove[grid,positions];

%t If[next=!=Missing[],AppendTo[positions,next],Break[];],{j,\[Infinity]}];

%t <|"Coordinates"->positions[[All,2]],"Indices"->positions[[All,1]]|>]

%t grid=ResourceFunction["LatticePointsArrangement"]["DiagonalZigZagEastQ4",10000];

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

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

%t fs

%t fs["Indices"]

%t ListPlot[fs["Indices"]]

%Y Cf. A316588, A316328, A316667.

%K nonn,fini,full,look

%O 1,2

%A _Sander G. Huisman_, Jan 28 2021

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Last modified May 11 09:02 EDT 2021. Contains 343784 sequences. (Running on oeis4.)