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

1,11

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).

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

Example

See representation of first points in Links section.

Crossrefs

Cf. A293539, A293730.

Sequence in context: A113780 A007515 A352455 * A319297 A309569 A356528

Adjacent sequences: A293726 A293727 A293728 * A293730 A293731 A293732

Keyword

sign,look

Author

Rémy Sigrist, Oct 15 2017

Status

approved