A293539 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 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/2, 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, 2, 3, 3, 2, 1, -1, -2, -2, 0, 1, 4, 4, 3, 2, 1, 0, -3, -3, -2, -1, 2, 5, 5, 4, 4, 5, 5, 4, 3, 2, 1, 0, -3, -4, -4, -3, -3, -4, -4, -3, -2, 0, -2, -3, -5, -5, -4, 0, 1, 1, -5, -5, -4, -4, -5, -6, -6

Offset: 1

Author

Rémy Sigrist, Oct 11 2017

Keywords

• sign
• look

Comments

See A293540 for the Y-coordinate of P(n).

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

. P(n)* .

. | t .

. | .

. | .

. |.

. P(n-1)*

. /

. /

. P(n-2)*

The sequence P has similarities with Langton's ant:

- after an apparently chaotic initial phase, an escape consisting of a repetitive pattern emerges at n = 9118 (see illustrations in Links section),

- more formally: P(n+258) = P(n) + (14,-8) for any n >= 9118,

- See A274369 and A274370 for the coordinates of Langton's ant,

- See also A293207 for other sequences of points with emerging escapes.

See also A292469 for a sequence of points with similar construction features.

Examples

			See representation of first points in Links section.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..12000
• Wikipedia, Langton's ant
• Rémy Sigrist, Representation of P(n) for n=1..42, with lines joining consecutive points
• Rémy Sigrist, Representation of P(n) for n=1..500, with lines joining consecutive points
• Rémy Sigrist, Representation of the repetitive pattern emerging at n=9118
• Rémy Sigrist, Colorized representation of the points P(n) for n=1..12000
• Rémy Sigrist, Colorized representation of the points P'(n) of the variant where we maximize the angle t in case of a tie for n=1..1000000
• Rémy Sigrist, PARI program for A293539

Crossrefs

Cf. A274369, A274370, A292469, A293207, A293540.

Programs

PARI

See Links section.

Formula

a(n + 258) = a(n) + 14 for any n >= 9118.