A293729 Let P be the sequence of distinct lattice points defined by the following rules: P(1) = (0,0), P(2) = (1,0), and for any n > 2, P(n) is the closest lattice point at integer distance to P(n-1) such that the angle of the vectors (P(n-2), P(n-1)) and (P(n-1), P(n)), say t, satisfies 0 < t < Pi, and in case of a tie, maximize the angle t; a(n) = X-coordinate of P(n).

0, 1, 1, 0, 0, 1, 1, 0, 0, 1, -3, -3, -2, -2, -4, -4, -1, -1, -5, -5, -2, -3, -3, -2, -2, -3, -3, -2, -2, -3, 1, 1, 0, 0, 2, 2, -1, -1, 3, 3, -1, -1, 0, 0, -2, -5, -4, -4, -5, -5, -4, -4, -5, -1, -1, -6, -6, -3, -4, -4, -3, -3, -4, -4, -1, -2, -2, -1, -1, -2

Offset: 1

Author

Rémy Sigrist, Oct 15 2017

Keywords

• sign
• look

Comments

See A293730 for the corresponding Y-coordinates.

The following diagram depicts the angle t cited in the name:

. P(n)* .

. | t .

. | .

. | .

. |.

. P(n-1)*

. /

. /

. P(n-2)*

This sequence has building features in common with A293539.

This sequence has fractal features: the representation of the first 200,000 points of the sequence seems chaotic; however, many portions of the sequence are duplicated elsewhere, possibly after rotating them by an integral multiple of 90 degrees (see representation of such duplication in Links section).

The variant P' of this sequence where we minimize the angle t in case of a tie has simple patterns that repeat themselves in a square spiral (see representation of P' in Links section).

Examples

			See representation of first points in Links section.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Representation of P(n) for n=1..100, with lines joining consecutive points
• Rémy Sigrist, Colorized representation of P(n) for n=1..200000
• Rémy Sigrist, Representation of P(n) with duplicate pattern highlighted
• Rémy Sigrist, Representation of the variant P'(n) for n=1..10000
• Rémy Sigrist, PARI program for A293729

Crossrefs

Cf. A293539, A293730.