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A307344 Cells visited by a single pawn move for an even cell and a double pawn move for an odd cell on a numbered 3D grid and moving to the lowest available unvisited cell of different parity at each step.

%I A307344 #9 Jul 09 2019 13:08:16
%S A307344 1,10,19,48,27,76,51,20,33,8,15,4,9,34,53,108,77,28,13,2,5,30,47,18,
%T A307344 31,6,3,12,23,72,49,102,71,22,11,36,21,70,101,186,131,252,193,106,75,
%U A307344 26,43,16
%N A307344 Cells visited by a single pawn move for an even cell and a double pawn move for an odd cell on a numbered 3D grid and moving to the lowest available unvisited cell of different parity at each step.
%C A307344 The grid is numbered as follows:
%C A307344    1: [0, 0, 0]
%C A307344    -- 1 step --
%C A307344    2: [0, 0, 1]
%C A307344    3: [0, 1, 0]
%C A307344    4: [1, 0, 0]
%C A307344    -- 2 steps --
%C A307344    5: [0, 0, 2]
%C A307344    6: [0, 1, 1]
%C A307344    7: [0, 2, 0]
%C A307344    8: [1, 0, 1]
%C A307344    9: [1, 1, 0]
%C A307344   10: [2, 0, 0]
%C A307344   etc.
%e A307344 1: [0, 0, 0] is an odd cell, hence a double move is required. Since 5: [0, 0, 2] and 7: [0, 2, 0] are also odd, 10: [2, 0, 0] is the only valid move.
%e A307344 The sequence ends at 16: [1, 1, 1]. A single move is required, which limits the possible destination cells to:
%e A307344    6: [0, 1, 1], even;
%e A307344    8: [1, 0, 1], even;
%e A307344    9: [1, 1, 0], already visited;
%e A307344   27: [1, 1, 2], already visited;
%e A307344   28: [1, 2, 1], even;
%e A307344   31: [2, 1, 1], already visited;
%Y A307344 Cf. A324273, A324274, A324275.
%K A307344 nonn,fini,full
%O A307344 1,2
%A A307344 _Jan Koornstra_, Apr 02 2019