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.

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

Author

Hugo Pfoertner, Mar 19 2019

Keywords

• sign
• look

Comments

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)

Links

• 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.

Crossrefs

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.

Programs

PARI

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)

Extensions

Name revised by Peter Munn, Jul 08 2021