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These are the numbers for which A000164(a(n)) = 4.

a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly four ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).

This sequence is a proper subsequence of A000378.

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.

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

More generally, the ordinary generating function for the number of ways of writing n as a sum of k squares is theta_3(0, q)^k = 1 + 2*k*q + 2*(k - 1)*k*q^2 + (4/3)*(k - 2)*(k - 1)*k*q^3 + (2/3)*((k - 3)*(k - 2)*(k - 1) + 3)*k*q^4 + (4/15) *(k - 1)*k*(k^3 - 9*k^2 + 26*k - 9)*q^5 + ..., where theta is the Jacobi theta functions.

These are the numbers for which A000164(a(n)) = 5.

a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly five ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).

This sequence is a proper subsequence of A000378.

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.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).