6th roots of unity

This page is about sequences that takes 6th root of unity as values.

The longest sequence with discrepancy 1 has length 116.

Method

Here should be a short description of the way the sequence was found. (The code(s) used should be further down this page.)

Status

Is the data still relevant (e.g. longest know)? Is the method still relevant, or have we found a better method? Is the program still running on a computer somewhere?

The data

If the [math]\displaystyle{ x_n }[/math] are allowed to be any of the six points of a regular hexagon, and one requires all sums along HAPs to be zero or one of those same points, the maximum length of a sequence is [math]\displaystyle{ 116 }[/math]. The following sequence achieves this, where the numbers index the points in order around the hexagon:

-, 0, 3, 3, 0, 3, 0, 2, 4, 0, 1, 4, 3,
1, 5, 0, 2, 4, 3, 1, 5, 5, 1, 4, 1, 2,
5, 3, 2, 5, 4, 1, 4, 1, 2, 5, 0, 2, 4,
4, 1, 5, 2, 1, 4, 3, 1, 5, 5, 2, 4, 1,
2, 4, 0, 1, 5, 4, 1, 4, 2, 2, 4, 0, 1,
5, 4, 2, 5, 1, 2, 4, 3, 1, 5, 5, 1, 4,
1, 3, 4, 2, 1, 0, 4, 1, 4, 3, 1, 5, 0,
2, 4, 5, 2, 4, 1, 1, 5, 3, 2, 0, 3, 4,
5, 1, 1, 3, 4, 1, 5, 0, 2, 4, 1, 3, 5

Here are some of its HAP subseqences, which seem to show the presence of some kind of multiplicative structure, though the structure seems to degenerate as the sequences progress.

The 2-sequence

3, 0, 0, 4, 1, 3, 5, 2, 3, 5, 1, 1, 5, 2, 4, 4, 2, 0, 4, 1, 2, 4, 1, 5, 4, 2, 0, 5, 1, 2, 4, 1, 4, 5, 2, 3, 5, 1, 1, 4, 1, 4, 4, 1, 0, 4, 2, 1, 5, 2, 3, 5, 1, 4, 5, 2, 1, 5

The 3-sequence

3, 0, 0, 3, 0, 3, 5, 1, 3, 4, 1, 0, 4, 2, 3, 5, 1, 0, 4, 2, 0, 4, 1, 3, 5, 1, 2, 4, 3, 0, 5, 1, 3, 3, 1, 4, 0, 1

The 4-sequence

0, 4, 3, 2, 5, 1, 2, 4, 0, 1, 4, 5, 2, 5, 2, 1, 5, 3, 1, 4, 4, 1, 4, 1, 2, 5, 4, 2, 5

The 5-sequence

3, 1, 0, 5, 2, 4, 5, 1, 3, 4, 1, 2, 5, 2, 5, 4, 1, 0, 4, 2, 1, 5, 3

The 6-sequence

0, 3, 3, 1, 4, 0, 2, 5, 0, 2, 4, 3, 1, 4, 0, 1, 3, 4, 1

The 8-sequence

4, 2, 1, 4, 1, 5, 5, 1, 3, 4, 1, 1, 5, 2

The 9-sequence

0, 3, 3, 0, 3, 0, 0, 3, 2, 0, 3, 4

The 10-sequence

1, 5, 4, 1, 4, 2, 2, 4, 0, 2, 5

The 12-sequence

3, 1, 0, 5, 2, 3, 4, 1, 4

Relevant code

The code(s) (or a link to the code(s)) used to find this sequence should be posted here.