

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(n1) such that the angle of the vectors (P(n2), P(n1)) and (P(n1), P(n)), say t, satisfies 0 < t <= Pi/2, and in case of a tie, minimize the angle t; a(n) = Xcoordinate 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
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OFFSET

1,8


COMMENTS

See A293540 for the Ycoordinate of P(n).
The following diagram depicts the angle t cited in the name:
. P(n)* .
.  t .
.  .
.  .
. .
. P(n1)*
. /
. /
. P(n2)*
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



