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%I #9 Sep 11 2021 19:28:10
%S 1,1,1,1,1,3,3,8,17,18,51,69,126,286,485,1180,2245,4312,7247,19559,
%T 41403,66454,161131,337495,710899,1286527,3069554,7100614,15046664,
%U 29090692,57797782,144576706,313441617,628544270,1278592587,3304149943,7192582862
%N a(n) is the number of solutions when placing the numbers 1..n in an n X n square according to the rules in the Comments section.
%C Place a 1 in the upper left corner of the square. Let k be the number placed last. At each step place k+1 at a distance k vertically or horizontally from k. Solutions are counted up to mirroring in the backward diagonal. That is, all solutions have 2 to the right of 1. Obviously, n is the largest possible value since the next step will inevitably lead outside the square.
%e Sample solutions:
%e n=6 n=7 n=8
%e 1 2 - - - - 1 2 - - - - - 1 2 - - - - 8 -
%e - - - - - - 7 5 - - - - 6 - - - - - - - -
%e - 3 - - - - - 3 - - - - - 5 3 - - 4 - - -
%e - - - - - - - - - - - - - - - - - - - - -
%e - - - - - - - - - - - - - - - - - - - - -
%e 6 4 - - - 5 - 4 - - - - - - - - - - - - -
%e - - - - - - - - - - - - - - -
%e 6 - - - - - 7 -
%Y Cf. A308353.
%K nonn
%O 1,6
%A _Lars Blomberg_, Nov 04 2019
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