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A292469 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 Z-coordinate of the cross product of the vectors (P(n-1), P(n)) and (P(n-1), P(j)) is strictly negative for j=1..n-2, and in case of a tie, P(n) maximizes the dot product of the vectors (P(n-2), P(n-1)) and (P(n-1), P(n)); a(n) = X-coordinate of P(n). 3

%I

%S 0,1,1,0,-1,-1,0,2,2,1,0,-1,-2,-2,-1,1,4,4,3,2,-1,-2,-3,-3,-2,-1,1,4,

%T 5,5,4,2,1,-1,-2,-3,-4,-4,-3,0,2,5,6,6,5,3,0,-1,-2,-3,-4,-5,-5,-4,-2,

%U 1,8,8,7,5,2,1,-2,-3,-4,-5,-6,-6,-5,-4,-2,1,5,10,10

%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 n > 2, P(n) is the closest lattice point to P(n-1) such that the Z-coordinate of the cross product of the vectors (P(n-1), P(n)) and (P(n-1), P(j)) is strictly negative for j=1..n-2, and in case of a tie, P(n) maximizes the dot product of the vectors (P(n-2), P(n-1)) and (P(n-1), P(n)); a(n) = X-coordinate of P(n).

%C More informally:

%C - the "scalar product" constraint means that the points P(1), ..., P(n-2) are all on the left side of the fixed vector (P(n-1), P(n)),

%C - the "dot product" constraint means the angle of the vectors (P(n-2), P(n-1)) and (P(n-1), P(n)) is maximized.

%C See A292470 for the Y-coordinate of P(n).

%C The points of sequence P spin around the origin, and the segments joining consecutive points of P do not intersect (except at the common endpoint of two consecutive segments); these properties are the original motivations for this sequence.

%H Rémy Sigrist, <a href="/A292469/b292469.txt">Table of n, a(n) for n = 1..1000</a>

%H Rémy Sigrist, <a href="/A292469/a292469.png">Representation of the first hundred points of P, with consecutive points joined by a segment</a>

%H Rémy Sigrist, <a href="/A292469/a292469_1.png">Representation of the first 500 points of P, with consecutive points joined by a segment</a>

%H Rémy Sigrist, <a href="/A292469/a292469.txt">C++ program for A292469</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross_product">Cross product</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dot_product">Dot product</a>

%e See representation of the first hundred points of P in Links section.

%o (C++) See Links section.

%Y Cf. A292470.

%K sign,look

%O 1,8

%A _Rémy Sigrist_, Sep 17 2017

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Last modified September 16 06:21 EDT 2019. Contains 327090 sequences. (Running on oeis4.)