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A353259 Solution to Snake Numbers Problems for Snakes from 1 to n for an n X n square grid (see Comments).

%I A353259 #17 Apr 10 2022 10:03:34
%S A353259 1,4,7,16,19,36,39,64,67,100
%N A353259 Solution to Snake Numbers Problems for Snakes from 1 to n for an n X n square grid (see Comments).
%C A353259 On an n X n square board we draw a path with squares that we number 1, 2, ..., n, 1, 2, ..., n, 1, ... of the greatest length, which will be a(n). The path advances from one number to the next horizontally or vertically, not diagonally, after n it goes back to number 1. The path cannot cross itself. There cannot be repeated numbers in the same row or column.
%H A353259 Rodolfo Kurchan, <a href="http://www.puzzlefun.online/problems"> Puzzle Fun, Snake Number Problem</a>.
%F A353259 a(n) = n^2 for even n.
%e A353259 a(5) = 19 from Giorgio Vecchi:
%e A353259   +---+---+---+---+---+
%e A353259   |  2|  3|  4|  5|  1|
%e A353259   +---+---+---+---+---+
%e A353259   |  1|   |   |  3|  2|
%e A353259   +---+---+---+---+---+
%e A353259   |   |  1|  5|  4|   |
%e A353259   +---+---+---+---+---+
%e A353259   |  3|  2|   |   |  4|
%e A353259   +---+---+---+---+---+
%e A353259   |  4|  5|  1|  2|  3|
%e A353259   +---+---+---+---+---+
%e A353259 .
%e A353259 a(7) = 39 from Giorgio Vecchi.
%e A353259 a(9) = 67 from Giorgio Vecchi.
%e A353259 For odd n >= 3 the solutions up to now have a(n) = (n-1)^2+3.
%K A353259 nonn,more
%O A353259 1,2
%A A353259 _Rodolfo Kurchan_, Apr 08 2022