A292469 - OEIS

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).

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(n-2) are all on the left side of the fixed vector (P(n-1), P(n)),

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

See A292470 for the Y-coordinate 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