

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(n1) such that the Zcoordinate of the cross product of the vectors (P(n1), P(n)) and (P(n1), P(j)) is strictly negative for j=1..n2, and in case of a tie, P(n) maximizes the dot product of the vectors (P(n2), P(n1)) and (P(n1), P(n)); a(n) = Xcoordinate of P(n).


3



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, 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, 1, 8, 8, 7, 5, 2, 1, 2, 3, 4, 5, 6, 6, 5, 4, 2, 1, 5, 10, 10
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,8


COMMENTS

More informally:
 the "scalar product" constraint means that the points P(1), ..., P(n2) are all on the left side of the fixed vector (P(n1), P(n)),
 the "dot product" constraint means the angle of the vectors (P(n2), P(n1)) and (P(n1), P(n)) is maximized.
See A292470 for the Ycoordinate of P(n).
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.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Representation of the first hundred points of P, with consecutive points joined by a segment
Rémy Sigrist, Representation of the first 500 points of P, with consecutive points joined by a segment
Rémy Sigrist, C++ program for A292469
Wikipedia, Cross product
Wikipedia, Dot product


EXAMPLE

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


PROG

(C++) See Links section.


CROSSREFS

Cf. A292470.
Sequence in context: A293772 A293680 A293539 * A307011 A285007 A194527
Adjacent sequences: A292466 A292467 A292468 * A292470 A292471 A292472


KEYWORD

sign,look


AUTHOR

Rémy Sigrist, Sep 17 2017


STATUS

approved



