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%I #23 Jan 20 2026 17:38:53
%S 1,1,1,8,2,8,6,24,128
%N Number of loops with length A389534(n) touching all cells in an n X n grid.
%C "Loop" means any continuous curve starting and ending at the same point.
%C "Touch" includes vertices and edges of grid cells.
%C Using the shortest known lengths: a(11) to a(13) >= 192, 12, 120 and a(6*k) >= 4^k/2 + 2^k for k >= 2, with equality iff k = 2. [Updated by _Tamás Fülöp_, Dec 14 2025]
%H Zoe Allen, <a href="https://math.stackexchange.com/q/5036912">Touching as many grid squares as possible with path of given length</a>, Math StackExchange, Feb 18 2025.
%H Tamás Fülöp, <a href="https://github.com/tomka700/Minimal-loop">C++ program</a>, GitHub repository, Oct 07 2025.
%H Tamás Fülöp, <a href="/A389534/a389534_1.png">All loops with n = 3 - 10</a>, Oct 07 2025.
%F a(0) = 1.
%F a(1) = oo.
%F a(n) <= 8 * A390109(n), if n != 1, with equality for all odd n >= 5.
%Y Cf. A389534, A390109.
%K nonn,more,hard
%O 2,4
%A _Tamás Fülöp_, Oct 25 2025