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%I #16 Oct 16 2017 09:57:07
%S 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,
%T 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,
%U -2,-1,1,2,3,0,0,1,2,3,4,5,5,4,3,1,-2,-7,-7,-6,-7
%N 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).
%C See A293681 for the corresponding Y-coordinates.
%C The following diagram depicts the angle t cited in the name:
%C . P(n)* .
%C . | t .
%C . | .
%C . | .
%C . |.
%C . P(n-1)*
%C . /
%C . /
%C . P(n-2)*
%C This sequence has building features in common with A293539.
%C 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.
%H Rémy Sigrist, <a href="/A293680/b293680.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A293680/a293680.png">Representation of P(n) for n=1..100, with lines joining consecutive points</a>
%H Rémy Sigrist, <a href="/A293680/a293680_1.png">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</a>
%H Rémy Sigrist, <a href="/A293680/a293680_2.png">Representation of P(n) for n=1..18698, with patterns repeated at least three times colored in red/green/blue</a>
%H Rémy Sigrist, <a href="/A293680/a293680.gp.txt">PARI program for A293680</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Langton%27s_ant">Langton's ant</a>
%e See representation of first points in Links section.
%o (PARI) See Links section.
%Y Cf. A293539, A293681.
%K sign,look
%O 1,8
%A _Rémy Sigrist_, Oct 14 2017
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