A305575 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

0, 1, 0, -1, 0, 1, -1, -1, 1, 2, 0, -2, 0, 2, 1, -1, -2, -2, -1, 1, 2, 2, -2, -2, 2, 3, 0, -3, 0, 3, 1, -1, -3, -3, -1, 1, 3, 3, 2, -2, -3, -3, -2, 2, 3, 4, 0, -4, 0, 4, 1, -1, -4, -4, -1, 1, 4, 3, -3, -3, 3, 4, 2, -2, -4, -4, -2, 2, 4, 5, 4, 3, 0, -3, -4, -5, -4, -3, 0, 3, 4, 5, 1, -1

Offset

0,10

Comments

Similar to A283307, but with secondary sorting by polar angle.

Links

Hugo Pfoertner, Table of n, a(n) for n = 0..17664 (covering range r <= 75).

Hugo Pfoertner, Illustration of A305576 vs A305575.

Rainer Rosenthal, Coloured Circular Visualization of Spiral [A305575,A305576]

Example

The first points (listing [polar angle phi,x,y]) are:
r^2
  0: [0.0*Pi,0,0];
  1: [0.0*Pi,1,0], [0.5*Pi,0,1], [1.0*Pi,-1,0], [1.5*Pi,0,-1];
  2: [0.25*Pi,1,1], [0.75*Pi,-1,1], [1.25*Pi,-1,-1], [1.75*Pi,1,-1];
  4: [0.0*Pi,2,0], [0.5*Pi,0,2], [1.0*Pi,-2,0], [1.5*Pi,0,-2];
  5: [0.148*Pi,2,1], [0.352*Pi,1,2], [0.648*Pi,-1,2], [0.852*Pi,-2,1],
   [1.148*Pi,-2,-1], [1.352*Pi,-1,-2], [1.648*Pi,1,-2], [1.852*Pi,2,-1];
  8: [0.25*Pi,2,2], [0.75*Pi,-2,2], [1.25*Pi,-2,-2], [1.75*Pi,2,-2].

Prog

(PARI) atan2(y, x)=if(x>0, atan(y/x), if(x==0, if(y>0, Pi/2, -Pi/2), if(y>=0, atan(y/x)+Pi, atan(y/x)-Pi)));
angle(x, y)=(atan2(y, x)+2*Pi)%(2*Pi);
{a004018(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))};
xyselect=1; \\ change to 2 for A305576
print1(0, ", "); for(s=1, 25, my(r=a004018(s)); if(r>0, my(v=matrix(r, 3), w=vector(r), m=sqrtint(s), L=0); for(i=-m, m, my(k=s-i^2, kk); if(k==0, v[L++, 1]=i; v[L, 2]=0; v[L, 3]=angle(i, 0), if(issquare(k), kk=sqrtint(k); forstep(j=-kk, kk, kk+kk, v[L++, 1]=i; v[L, 2]=j; v[L, 3]=angle(i, j))))); p=vecsort(v[, 3], , 1); for(k=1, L, w[k]=v[p[k], xyselect]); for(k=1, L, print1(w[k], ", ")))); \\ Hugo Pfoertner, May 12 2019

Crossrefs

For the y-coordinates see A305576.

Cf. A000328, A004018, A283307, A283308, A307014, A307016, A307017, A308080, A308081.

Sequence in context: A054533 A339927 A227957 * A247977 A359239 A143232

Adjacent sequences: A305572 A305573 A305574 * A305576 A305577 A305578

Keyword

sign,look

Author

Hugo Pfoertner, Jun 05 2018

Status

approved