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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
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.
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
KEYWORD
sign,look
AUTHOR
Rémy Sigrist, Oct 11 2017
STATUS
approved