SuneJ: 6th unity roots moved to 6th roots of unity

2010-01-09T19:01:18Z

6th unity roots moved to 6th roots of unity

← Older revision	Revision as of 12:01, 9 January 2010
(No difference)

SuneJ: New page: 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 th...

2010-01-09T18:58:55Z

New page: 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 th...

New page

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>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>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.

6th roots of unity - Revision history

SuneJ: 6th unity roots moved to 6th roots of unity

SuneJ: New page: 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 th...