Equal Distances

Points with the same distance are taken in anti-clockwise order around from the X axis. For example X=3,Y=1 is sqrt(10) from the origin, as are the swapped X=1,Y=3, and X=-1,Y=3 etc in other quadrants, for a total 8 points N=30 to N=37 all the same distance.

When one of X or Y is 0 there's no negative, so just four negations like N=10 to 13 points X=2,Y=0 through X=0,Y=-2. Or on the diagonal X==Y there's no swap, so just four like N=22 to N=25 points X=3,Y=3 through X=3,Y=-3.

There can be more than one way for the same distance to arise. A Pythagorean triple like 3^2 + 4^2 == 5^2 has 8 points from the 3,4, then 4 points from the 5,0 giving a total 12 points N=70 to N=81. Other combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and also with more than two different ways to have the same sum.

Multiples of 4

The first point of a given distance from the origin is either on the X axis or somewhere in the first octant. The row Y=1 just above the axis is the first of its equals from X>=2 onwards, and similarly further rows for big enough X.

There's always a multiple of 4 many points with the same distance so the first point has N=4*k+2, and similarly on the negative X side N=4*j, for some k or j. If you plot the prime numbers on the path then those even N's (composites) are gaps just above the positive X axis, and on or just below the negative X axis.

Circle Lattice

Gauss's circle lattice problem asks how many integer X,Y points there are within a circle of radius R.

The points on the X axis N=2,10,26,46, etc are the first for which X^2+Y^2==R^2 (integer X==R). Adding option "n_start=>0" to make them each 1 less gives the number of points strictly inside, ie. X^2+Y^2 < R^2.

The last point satisfying X^2+Y^2==R^2 is either in the octant below the X axis, or is on the negative Y axis. Those N's are the number of points X^2+Y^2<=R^2, Sloane's A000328.

When that A000328 sequence is plotted on the path a straight line can be seen in the fourth quadrant extending down just above the diagonal. It arises from multiples of the Pythagorean 3^2 + 4^2, first X=4,Y=-3, then X=8,Y=-6, etc X=4*k,Y=-3*k. But sometimes the multiple is not the last among those of that 5*k radius, so there's gaps in the line. For example 20,-15 is not the last since because 24,-7 is also 25 away from the origin.

Even Points

Option "points => "even"" visits just the even points, meaning the sum X+Y even, so X,Y both even or both odd.

    points => "even"
          52    40    39    51             5
       47    32    23    31    46          4
    53    27    16    15    26    50       3
       33    11     7    10    30          2
    41    17     3     2    14    38       1
       24     8     1     6    22     <- Y=0
    42    18     4     5    21    45      -1
       34    12     9    13    37         -2
    54    28    19    20    29    57      -3
       48    35    25    36    49         -4
          55    43    44    56            -5
                    ^
    -5 -4 -3 -2 -1 X=0 1  2  3  4  5

Even points can be mapped to all points by a 45 degree rotate and flip. N=1,6,22,etc on the X axis here is on the X=Y diagonal of all-points. And conversely N=1,2,10,26,etc on the X=Y diagonal here is the X axis of all-points.

The sets of points with equal hypotenuse are the same in the even and all, but the flip takes them in a reversed order.

Odd Points

Option "points => "odd"" visits just the odd points, meaning sum X+Y odd, so X,Y one odd the other even.

    points => "odd"
                                             
             71    55    54    70                6
          63    47    36    46    62             5  
       64    37    27    26    35    61          4  
    72    38    19    14    18    34    69       3  
       48    20     7     6    17    45          2  
    56    28     8     2     5    25    53       1  
       39    15     3  +  1    13    33     <- Y=0  
    57    29     9     4    12    32    60      -1  
       49    21    10    11    24    52         -2  
    73    40    22    16    23    44    76      -3  
       65    41    30    31    43    68         -4  
          66    50    42    51    67            -5  
             74    58    59    75               -6
                                             
                       ^
    -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

Odd points can be mapped to all points by a 45 degree rotate and a shift X-1,Y+1 to put N=1 at the origin. The effect of that shift is as if the hypot measure in "all" points was (X-1/2)^2+(Y-1/2)^2 and for that reason the sets of points with equal hypots are not the same in odd and all.