OFFSET
0,2
COMMENTS
Board is numbered with the square spiral:
.
16--15--14--13--12
| |
17 4---3---2 11 .
| | | |
18 5 0---1 10 .
| | |
19 6---7---8---9 .
|
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
KEYWORD
nonn,fini,full
AUTHOR
Daniël Karssen, Aug 01 2018
STATUS
approved