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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). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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.

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

FORMULA

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

EXAMPLE

See representation of first points in Links section.

PROG

(PARI) See Links section.

CROSSREFS

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

Sequence in context: A055892 A293772 A293680 * A292469 A307011 A285007

Adjacent sequences:  A293536 A293537 A293538 * A293540 A293541 A293542

KEYWORD

sign,look

AUTHOR

Rémy Sigrist, Oct 11 2017

STATUS

approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)