A293680 - OEIS

A293680

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 i < j < k, P(k) does not lie on the vector (P(i), P(j)), and for any n > 2, P(n) is the closest lattice point 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, minimize the angle t; a(n) = X-coordinate of P(n).

0, 1, 1, 0, -1, -1, 0, 2, 2, 1, 0, -1, -2, -2, -1, 1, 3, 3, 2, 1, -1, -2, -3, -3, -2, -1, 1, 2, 4, 4, 3, 2, 0, -3, -4, -4, -3, -2, -1, 1, 2, 3, 4, 4, 3, 1, -2, -5, -5, -4, -5, -6, -6, -5, -4, -2, -1, 1, 2, 3, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 1, -2, -7, -7, -6, -7

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

See A293681 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.

The study of the first thousand dots shows an alternation of apparently chaotic phases and regular phases where a pattern repeats itself; unlike Langton's ant, this repetitive behavior doesn't last long. It is unknown if eventually a periodic pattern repeating itself infinitely emerges.

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, Representation of P(n) for n=1..1000, with lines joining consecutive points and patterns repeated at least three times colored in red/green/blue

Rémy Sigrist, Representation of P(n) for n=1..18698, with patterns repeated at least three times colored in red/green/blue

Rémy Sigrist, PARI program for A293680

Wikipedia, Langton's ant

EXAMPLE

See representation of first points in Links section.

PROG

(PARI) See Links section.

CROSSREFS