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

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

Cf. A293539, A293681.

Sequence in context: A015504 A055892 A293772 * A293539 A292469 A357701

Adjacent sequences: A293677 A293678 A293679 * A293681 A293682 A293683

Keyword

sign,look

Author

Rémy Sigrist, Oct 14 2017

Status

approved