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

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

Offset: 0

Author

Hugo Pfoertner, Mar 19 2019

Keywords

• sign
• look

Comments

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 1. 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 2 has positive distance. - Peter Munn, Jul 13 2021

We can use this coordinate with the first coordinate to form an oblique coordinate system, in which each coordinate maps to an oblique coordinate vector parallel to the axis along which the other coordinate is 0. See the figure with nonperpendicular axes in the Barile link. When both of these coordinates are positive, the oblique coordinate vectors make a 60-degree angle with each other. [Made more specific by Peter Munn, Jul 19 2021]

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..10150
• Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
• HandWiki, Hexagonal Lattice.
• Peter Munn, Illustration of signed distance of spiral points.
• Wikipedia, Signed distance function.
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A307011, A307013, A328818, A334493, A345435, A345978.

Extensions

Name revised by Peter Munn, Jul 08 2021