A366153 Starting with the n-th shortest Cartesian line segment, a(n) is the minimal number of consecutive line segments required to make a simple polygon.

4, 5, 4, 3, 4, 3, 4, 4, 5, 4, 5, 4, 4, 6, 5, 6, 5, 6, 5, 7, 6, 4, 5, 5, 4, 4, 5

Offset

1,1

Comments

List the possible lengths of line segments achievable by connecting integral coordinates on a Cartesian grid. Starting from the n-th length, a(n) is the smallest number of consecutively greater lengths required to form a simple polygon with all vertices on integral Cartesian coordinates.

Links

Table of n, a(n) for n=1..27.

Gordon Hamilton, Rooting for NASA.

Example

a(4) = 3 because i) the fourth, fifth and sixth lengths are sqrt(5), sqrt(8) and 3 and ii) a triangle can be created using edges with these three lengths.
a(5) = 4 because i) the fifth, sixth, seventh and eighth lengths are sqrt(8), 3, sqrt(10), sqrt(13) and ii) a quadrilateral can be created using edges with these four lengths and iii) the fifth, sixth and seventh lengths alone cannot create a simple polygon with integral Cartesian vertices.

Crossrefs

Cf. A001481 (List of the squares of possible line segment lengths with both endpoints integral Cartesian coordinates).

Sequence in context: A071413 A161811 A016717 * A031349 A200605 A036444

Adjacent sequences: A366150 A366151 A366152 * A366154 A366155 A366156

Keyword

nonn,more

Author

Gordon Hamilton, Sep 28 2023

Status

approved