A317619 Squares visited by a (2,4)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step.

0, 50, 12, 18, 24, 14, 20, 10, 16, 22, 52, 58, 128, 134, 60, 54, 124, 78, 160, 70, 76, 158, 152, 66, 140, 146, 68, 62, 136, 130, 232, 122, 164, 74, 156, 266, 72, 150, 64, 138, 56, 126, 80, 162, 276, 154, 264, 406, 144, 250, 388, 242, 236, 370, 228, 166

Offset

0,2

Comments

Board is numbered with the square spiral:

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16--15--14--13--12

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17 4---3---2 11 .

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18 5 0---1 10 .

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19 6---7---8---9 .

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20--21--22--23--24--25

.

The sequence is finite: at step 2015, square 8398 is visited, after which there are no unvisited squares within one move.

Links

Daniël Karssen, Table of n, a(n) for n = 0..2015

Daniël Karssen, Figure showing the first 29 steps of the sequence

Daniël Karssen, Figure showing the complete sequence

Formula

a(n) = A317620(n+1) - 1.

Mathematica

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 = 0, 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 *)

Crossrefs

Cf. A317620, A317621.

Sequence in context: A187526 A187697 A048178 * A330848 A072460 A248128

Adjacent sequences: A317616 A317617 A317618 * A317620 A317621 A317622

Keyword

nonn,fini,full

Author

Daniël Karssen, Aug 01 2018

Status

approved