A307013 Third coordinate (asymmetric variant) in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and second coordinates are given in A307011 and A307012.

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

Offset: 0

Author

Hugo Pfoertner, Mar 19 2019

Keywords

• sign
• look

Comments

From Peter Munn, Jul 11 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 three "0-axes" that pass through spiral point 0 and one of points 1, 2 or 3. These 0-axes are the lines along which one of the coordinates is 0.

a(n), the 3rd coordinate, is the signed distance from spiral point n to the coordinate's 0-axis, which passes through points 0 and 3. 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 3rd coordinate is the sum of the other 2. In the symmetric variant of the coordinate system, the 3rd coordinate has the opposite sense, so that the 3 coordinates sum to 0. See A345978.

We can use any 2 of the 3 coordinates to form an oblique coordinate system, in which each of the 2 coordinates specifies vectors parallel to the other coordinate's 0-axis. This means the direction of the oblique coordinate vectors depends on the choice of the other coordinate - see the illustration of coordinate pairing in the links. When both coordinates are positive, an oblique coordinate vector derived from this sequence makes a 120-degree angle with the vector derived from the other sequence; however, when A307011 and A307012 are used together, the angle is 60 degrees.

Pairing with A307012 can be viewed as follows. Let omega = -1/2 + i*sqrt(3)/2, a primitive cube root of unity. Then f(n) = a(n) + omega*A307012(n) embeds the spiral in the complex plane with spiral points 0 and 1 embedded at 0 and 1 (so that the points of the spiral embed as the Eisenstein integers, as used for A345435).

(End)

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..10092
• Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
• HandWiki, Hexagonal Lattice.
• Peter Munn, Illustration of coordinate pairing.
• Eric Weisstein's World of Mathematics, Eisenstein Integer
• Wikipedia, Signed distance function.

Crossrefs

Cf. A307011, A307012, A328818, A345435, A345978.

A334492 is effectively this "3rd coordinate" for a different sequence of points on a hexagonal lattice.

Formula

a(n) = A307011(n) + A307012(n). - Peter Munn, Jul 04 2021