A307017 -id:A307017 - cp's OEIS frontend

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

Hugo Pfoertner, Jun 05 2018

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

			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].

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]

For the y-coordinates see A305576.

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

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

A305576 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 y-coordinates.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jun 05 2018

Similar to A283308, but with secondary sorting by polar angle.

			See A305575.

Rainer Rosenthal, Table of n, a(n) for n = 0..17664, covering range r <= 75 (terms 0..7844 from Hugo Pfoertner).

For the x-coordinates see A305575.

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

PARI
```
\\ Provided in A305575.
```

A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 22 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.
a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.
This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.
There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10034
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Hugo Pfoertner, Illustration of A307012 vs A307011, spiral.
Hugo Pfoertner, Illustration of A345978 vs A307011, spiral.
Wikipedia, Signed distance function.

Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978.
Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451.
Positions on the spiral that correspond to Eisenstein primes: A345435.

r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d)

Name revised by Peter Munn, Jul 08 2021

A307014 List coordinates (x,y) of the points in an hexagonal grid, 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 the first coordinate in a barycentric coordinate system.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Mar 21 2019

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives i. j is given in A307016, k is given in A307017.

The sorting by polar angle affects the grid points in the shells of size A035019, starting at indices given by A038590.

Hugo Pfoertner, Table of n, a(n) for n = 0..9060, covering range r <= 50.
Hugo Pfoertner, Illustration of initial terms of A307014 and A307016.
Hugo Pfoertner, Illustration of A307016 vs A307014, grid points.
Hugo Pfoertner, Illustration of A307016 vs A307014, spiral.
Hugo Pfoertner, Illustration of A307017 vs A307014, spiral.
Hugo Pfoertner, PARI program for A307014 and A307016.

Cf. A004016, A035019, A038590, A305575, A305576, A307011, A307012, A307013, A307016, A307017.

\\ See Link
\\ To create the data of this sequence load program from file and call
a307014_16(5, 4) \\ Hugo Pfoertner, Nov 07 2023

A307016 List coordinates (x,y) of the points in an hexagonal grid, 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 the second coordinate in a barycentric coordinate system.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Mar 21 2019

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives j. i is given in A307014, k is given in A307017.

Hugo Pfoertner, Table of n, a(n) for n = 0..9060, covering range r <= 50.
Hugo Pfoertner, PARI program for A307014 and A307016.

Cf. A004016, A035019, A038590, A307014, A307017.

\\ See Link
\\ To create the data of this sequence load program from file and call
a307014_16(5, 6) \\ Hugo Pfoertner, Nov 07 2023

cp's OEIS Frontend

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.

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A305576 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 y-coordinates.

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A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

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A307014 List coordinates (x,y) of the points in an hexagonal grid, 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 the first coordinate in a barycentric coordinate system.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A307016 List coordinates (x,y) of the points in an hexagonal grid, 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 the second coordinate in a barycentric coordinate system.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI