A342561 List points (x,y,z) having integer coordinates, sorted first by R^2 = x^2 + y^2 + z^2 and in case of ties, then by z and last by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 1, -1, -1, 1, 1, 0, -1, 0, 1, -1, -1, 1, 1, -1, -1, 1, 0, 2, 0, -2, 0, 0, 1, 0, -1, 0, 2, 0, -2, 0, 2, 1, -1, -2, -2, -1, 1, 2, 2, 0, -2, 0, 1, 0, -1, 0, 1, -1, -1, 1, 2, 1, -1, -2, -2, -1, 1, 2, 2, 1, -1, -2, -2, -1, 1, 2, 1, -1, -1, 1, 2, 0, -2, 0, 2, -2, -2, 2, 2, 0, -2, 0

Offset: 0

Author

Hugo Pfoertner, Apr 27 2021

Keywords

• sign

Comments

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

Examples

			   n    x    y    z  R^2  phi/Pi
   0    0    0    0   0   0.000
   1    0    0   -1   1   0.000
   2    1    0    0   1   0.000
   3    0    1    0   1   0.500
   4   -1    0    0   1   1.000
   5    0   -1    0   1   1.500
   6    0    0    1   1   0.000
   7    1    0   -1   2   0.000
   8    0    1   -1   2   0.500
   9   -1    0   -1   2   1.000
  10    0   -1   -1   2   1.500
  11    1    1    0   2   0.250
  12   -1    1    0   2   0.750
  13   -1   -1    0   2   1.250
  14    1   -1    0   2   1.750
  15    1    0    1   2   0.000
  16    0    1    1   2   0.500
  17   -1    0    1   2   1.000
  18    0   -1    1   2   1.500
  19    1    1   -1   3   0.250
  20   -1    1   -1   3   0.750
  21   -1   -1   -1   3   1.250
  22    1   -1   -1   3   1.750
  23    1    1    1   3   0.250
  24   -1    1    1   3   0.750
  25   -1   -1    1   3   1.250
  26    1   -1    1   3   1.750
  27    0    0   -2   4   0.000
  28    2    0    0   4   0.000
  29    0    2    0   4   0.500
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..10130

Crossrefs

Cf. A005875, A117609, A305575, A305576, A342562, A342563.

Cf. A343630, A340631, A340632, A343633 for a variant which "connects" corresponding poles of successive shells, A343640, A340641, A340642, A343643 for a square spiral variant.

Programs

PARI

shell(n, Q=Qfb(1,0,1), L=List())={for(z=if(n, sqrtint((n-1)\3)+1), sqrtint(n), my(S=if(n>z^2, Set(apply(vecsort, abs(qfbsolve(Q, n-z^2, 3)))), [[0,0]])); foreach(S, s, forperm(concat(s,z), p, listput(L, p)))); for(i=1,3, for(j=1,#L, my(X=L[j]); (X[i]*=-1) && listput(L,X))); vecsort(L, (p,q)->if( p[3]!=q[3], p[3]-q[3], p[1]==q[1], q[2]-p[2], p[2]*q[2]<0, q[2]-p[2], (q[1]-p[1])*(p[2]+q[2])))} \\ Gives list of all points with Euclidean norm sqrt(n).
A342561_vec=concat([[P[1] | P <- shell(n)] | n<-[0..7]]) \\ M. F. Hasler, Apr 27 2021