A343746 The x,y,z coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

0, 0, 0, 0, 1, 2, 0, -1, 1, 0, 0, -1, 0, 1, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, -1, -1, 2, -1, -2, 1, -1, 0, -1, 0, 0, -1, 1, 2, 1, 2, 2, 2, 2, 0, 1, 0, 0, -1, -1, 0, -1, -2, 2, 1, -2, 3, 2, 0, 3, 2, -2, 2, 3, 0, 2, 3, -2, 1, 2, -2, 3, 2, -3, 1, 0, -4, 1, -2, -3, 1, -2, -2, 3, -3, 0, 3, -2, 2

Offset: 1

Author

Scott R. Shannon, Apr 27 2021

Keywords

• sign
• fini

Comments

See A343678 for the rules determining the knight's steps on the cubic lattice and for images of the resulting path.

Examples

			a(1),a(2),a(3) = 0,0,0. The knight starts at the origin on the lattice.
a(4),a(5),a(6) = 0,1,2. The 24 points the knight could step to on the first step all have only 1 visited neighbor and are all the same distance from the origin. Also they all have coordinates as arrangements of 0,+-1,+-2 thus the only way they can be separated is using rules 5 and 6 of A343678 which selects the smallest magnitudes of the x,y,z coordinates followed by the largest absolute x,y,z coordinates. This leads to the point (0,1,2) being selected.
a(7),a(8),a(9) = 0,-1,1. The other two possible points which have the same number of visited neighbors and are the same distance from the origin are (1,1,0) and (-1,1,0), but (0,-1,1) is chosen as that has the minimum x-coordinate magnitude.
a(52),a(53),a(54) = -1,-2,2. This is the first point that is chosen due to having the maximum product of the absolute values of its coordinate.
a(577),a(578),a(579) = -2,-3,-1. This is the first point that is stepped to that has two visited neighboring points.
a(1978),a(1979),a(1980) = -3,3,7. This is the first point that is stepped to that has three visited neighboring points.
		

Crossrefs

Cf. A343678 (point square distances from origin), A343747 (point x coordinates), A343748 (point y coordinates), A343749 (point z coordinates) A330189, A329520, A316667.