A324275 -id:A324275 - cp's OEIS frontend

A324274 a(n) is the number of squares visited by a single pawn move for an even square and a double pawn move for an odd square on a diagonally numbered board and moving to the lowest available unvisited square of different parity at each step from subsequent starting squares n; or a(n) = 0 for an infinite length.

20, 0, 8, 17, 6, 9, 4, 7, 0, 11, 16, 5, 18, 0, 10, 19, 8, 19, 8, 11, 12, 25, 6, 9, 6, 9, 0, 13, 24, 7, 20, 7, 20, 0, 12, 15, 24, 21, 26, 21, 10, 21, 10, 13, 14, 27, 8, 27, 8, 11, 8, 11, 0, 15, 16, 33, 22, 9, 22, 9, 22, 9, 22, 0, 14, 17, 32, 23

Offset: 1

Jan Koornstra, Feb 20 2019

nonn

It is conjectured that all starting squares will either have a finite length or reach the top row of the board at square 2 first and then follow the sequence for a(2) to infinity. A324275 contains numbers n for which A324274(n) = 0.

			a(1) is the length of A324273. a(2) has an infinite length as it will follow a repeating pattern along the top row of the numbered board.

Cf. A316588, A324273, A324275.

A324273 Squares visited by a single pawn move for an even square and a double pawn move for an odd square on a diagonally numbered board and moving to the lowest available unvisited square of different parity at each step.

Original entry on oeis.org

1, 4, 7, 2, 5, 12, 17, 8, 13, 6, 3, 10, 15, 26, 19, 32, 25, 14, 9, 18

Offset: 1

Jan Koornstra, Feb 20 2019

The board is numbered as follows:
2  4  7 11 16  .
5  8 12 17  .
9 13 18  .
14 19  .
20  .
.
   .

			Square 1 is odd. Hence the next square should be the lowest even square a double move away from 1, which is square 4. Next, there is only a single option to move in a single move to an odd square, namely at square 7. Etc.

Cf. A316588, A324274, A324275.

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.

Original entry on oeis.org

1, 10, 19, 48, 27, 76, 51, 20, 33, 8, 15, 4, 9, 34, 53, 108, 77, 28, 13, 2, 5, 30, 47, 18, 31, 6, 3, 12, 23, 72, 49, 102, 71, 22, 11, 36, 21, 70, 101, 186, 131, 252, 193, 106, 75, 26, 43, 16

Offset: 1

Jan Koornstra, Apr 02 2019

The grid is numbered as follows:
[0, 0, 0]
   -- 1 step --
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
   -- 2 steps --
[0, 0, 2]
[0, 1, 1]
[0, 2, 0]
[1, 0, 1]
[1, 1, 0]
[2, 0, 0]
  etc.

			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.
The sequence ends at 16: [1, 1, 1]. A single move is required, which limits the possible destination cells to:
   6: [0, 1, 1], even;
   8: [1, 0, 1], even;
   9: [1, 1, 0], already visited;
  27: [1, 1, 2], already visited;
  28: [1, 2, 1], even;
  31: [2, 1, 1], already visited;

Cf. A324273, A324274, A324275.

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A324273 Squares visited by a single pawn move for an even square and a double pawn move for an odd square on a diagonally numbered board and moving to the lowest available unvisited square of different parity at each step.

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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.

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