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Squares visited by a (2,4)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step, squares labelled >=1.
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%I #16 Feb 06 2025 04:13:56

%S 1,51,13,19,25,15,21,11,17,23,53,59,129,135,61,55,125,79,161,71,77,

%T 159,153,67,141,147,69,63,137,131,233,123,165,75,157,267,73,151,65,

%U 139,57,127,81,163,277,155,265,407,145,251,389,243,237,371,229,167

%N Squares visited by a (2,4)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step, squares labelled >=1.

%C Board is numbered with the square spiral:

%C .

%C 17--16--15--14--13

%C | |

%C 18 5---4---3 12 .

%C | | | |

%C 19 6 1---2 11 .

%C | | |

%C 20 7---8---9--10 .

%C |

%C 21--22--23--24--25--26

%C .

%C The sequence is finite: at step 2016, square 8399 is visited, after which there are no unvisited squares within one move.

%H Daniël Karssen, <a href="/A317620/b317620.txt">Table of n, a(n) for n = 1..2016</a>

%H Daniël Karssen, <a href="/A317620/a317620.svg">Figure showing the first 29 steps of the sequence</a>

%H Daniël Karssen, <a href="/A317620/a317620_1.svg">Figure showing the complete sequence</a>

%F a(n) = A317619(n-1) + 1.

%t spiral[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; Block[{nn = 30, j = {2, 4}, k = 1, t}, t = spiral[nn]; (k - 1) + Nest[Function[{a}, Append[a, SelectFirst[Sort@ Map[{t[[##]], ##} & @@ {#1 + a[[-1, 2]], #2 + a[[-1, 3]]} & @@ # &, Join @@ Inner[Times, Tuples[{-1, 1}, {2}], Permutations[j], List]], FreeQ[a[[All, 1]], First[#] ] &]]], {{1, nn, nn}}, 55][[All, 1]] ] (* _Michael De Vlieger_, Aug 02 2018 *)

%Y Cf. A317619, A317622.

%K nonn,fini,full

%O 1,2

%A _Daniël Karssen_, Aug 01 2018