A324273 -id:A324273 - 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.

A324275 Numbers k for which A324274(k) is 0, i.e., starting squares in A324274 that yield a path of infinite length.

Original entry on oeis.org

2, 9, 14, 27, 34, 53, 64, 89, 102, 133, 150, 187, 206, 249, 272, 321, 346, 401, 430, 491, 522, 589, 624, 697, 734, 813, 854, 939, 982, 1073, 1120, 1217, 1266, 1369, 1422, 1531, 1586, 1701, 1760, 1881, 1942, 2069, 2134, 2267, 2334, 2473

Offset: 1

Jan Koornstra, Feb 27 2019

nonn

Note that the sequence up to a(n) (for its current known values) is actually the path of a(n) in reverse until it reaches square 2. It is therefore conjectured that all starting squares in A324274 either have a finite length or are part of this single sequence.

Cf. A316588, A324273, A324274.

Conjectures from Colin Barker, Mar 09 2019: (Start)
G.f.: x*(2 + 7*x + 3*x^2 + 6*x^3 - x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).
a(n) = (5 + 7*(-1)^n + (2-2*i)*(-i)^n + (2+2*i)*i^n + (26+6*(-1)^n)*n + 18*n^2) / 16 where i=sqrt(-1).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>7.
(End)

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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A324275 Numbers k for which A324274(k) is 0, i.e., starting squares in A324274 that yield a path of infinite length.

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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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