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A342561
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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.
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7
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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
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OFFSET
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0,29
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COMMENTS
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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
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LINKS
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EXAMPLE
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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
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PROG
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(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).
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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