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This is a 3D generalization of the 2D square spiral and could be used to produce a 3D variant of Ulam's prime spiral.

See A343630 for an analog using the Euclidean or 2-norm instead of the sup- or oo-norm used here, so points are partitioned in spheres and circles instead of squares and cubes here.

The integer lattice points, Z^3, are listed in order of increasing sup norm R = max(|x|, |y|, |z|). Each "sphere" or shell of given radius R is filled starting at the North or South pole using concentric squares on the top and bottom face and squares of fixed size (2R+1) X (2R+1) at intermediate z-coordinates. Each square (circle for the sup-norm) is filled in the sense of increasing longitude, where the positive x axis corresponds to longitude 0, i.e., the points (r,0,z), (0,r,z), (-r,0,z) and (0,-r,z) are visited in this order. The z-values are alternatively increasing and decreasing (so over a period of two shells they follow the same rectangle-wave shape as the x-values do over the period of each square).

The sequence can be seen as a table with row length of 3, where each row corresponds to the (x,y,z)-coordinates of one point (then the three columns are A343641, A343642 and A343643), or as a table with row lengths 3*A010014, where A010014(r) is the number of points with sup-norm r.

There are (2n+1)^3 integer lattice points with sup norm <= n. Therefore, the point number n (where 0 is the origin) is in the shell r = round(n^(1/3)/2) = floor(...+1/2). Within shell r, which starts with the point number (2r-1)^3 (except for r=0), the first and last (2r+1)^2 points are on square spirals on the top and bottom faces, and the other points are on 2r-1 squares forming "belts" of 8r points each, on the side faces of the cube.

This is a 3-dimensional generalization of A305575 and A305576.

y-coordinates are in A342562, z-coordinates are in A342563.

These lists can be read as an irregular table, where row r lists the respective coordinates of the points on the sphere with radius R = sqrt(r); their number (i.e., the row length) is given by A005875 = (1, 6, 12, 8, 6, 24, 24, 0, 12, 30, ...). - M. F. Hasler, Apr 27 2021

See the main entry A343630 for details about this 3D generalization of an Ulam type spiral using the Euclidean norm.

Sequences A343631 and A343632 give the x and y-coordinates.

The sequence can be seen as a table with row lengths A005875, where A005875(r) is the number of points at distance sqrt(r) from the origin.

Sequence A343643 is the analog for the square spiral variant A343640.

See A343640 for more information about this 3D generalization of the 2D Ulam type square spiral.

The sequence can be seen as a table with row lengths A010014, where A010014(r) is the number of points of Z^3 with sup-norm r.

The graph of this sequence oscillates with increasing amplitude and wave length.

This is a 3D generalization of a plane filling spiral using the Euclidean norm.

See A343640 for an analog using the sup- or oo-norm, where circles are squares and spheres are cubes.

The integer lattice points, Z^3, are listed in order of increasing Euclidean distance R^2 = x^2 + y^2 + z^2 from the origin. Each shell of given radius is filled using circles located at given latitude (i.e., z-value) on the sphere, and each circle is filled by points with increasing longitude, where the positive x axis corresponds to longitude 0. The latitudes / z-values are alternately increasing and decreasing (so over a period of two shells they follow the same cosine-type shape as the x-values do over the period of each circle).

The sequence can be seen as a table with row length of 3, where each row corresponds to the (x,y,z)-coordinates of one point (the three columns are then A343631, A343632 and A343633), or as a table with row lengths 3*A005875, where A005875(r) is the number of points at distance sqrt(r) from the origin.

Sequence A343640 gives a square spiral variant.