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[[The first 1124-sequence]] with discrepancy 2. ''Some more description''
* [[The first 1124-sequence]] with discrepancy 2. ''Some more description''
 
* Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''
Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''
* Some data about the problem with [[different upper and lower bound]]. ''Some more description''
 
* Sequences taking values in <math>\mathbb{T}</math>:
Some data about the problem with [[different upper and lower bound]]. ''Some more description''
** [[4th roots of unity]]
 
** [[6th roots of unity]]
 
 
== Geometric variations ==
 
It has been pointed out that the problem can be generalized to higher dimensions, for example by considering sequences with <math>\Vert x_n \Vert_2 = 1</math> having partial sums lying within a sphere. It is difficult to do much in the way of computation when the <math>x_n</math> can vary continuously, but if they are restricted to some finite set the problem becomes purely combinatorial and one can do more.
 
[[6th unity roots]]
 
=== The nine-point square ===
 
If the <math>x_n</math> are allowed to be any of the four points <math>(\pm 1, 0)</math> and <math>(0, \pm 1)</math>, and one requires all sums along HAPs to belong to one of the nine points at unit spacing centred on the origin, the maximum length of a sequence is ''at least'' <math>314</math>. The following sequence achieves this:
 
(1, 0), (0, 1), (-1, 0), (1, 0), (-1, 0), (0, -1), (0, 1), (-1, 0), (1, 0), (1, 0), (0, -1), (-1, 0), (-1, 0), (0, -1), (1, 0), (1, 0), (-1, 0), (0, 1), (1, 0), (-1, 0), (0, -1), (-1, 0), (0, 1), (1, 0), (0, -1), (1, 0), (-1, 0), (0, 1), (1, 0), (-1, 0), (0, -1), (-1, 0), (0, 1), (0, -1), (1, 0), (1, 0), (-1, 0), (-1, 0), (1, 0), (1, 0), (0, 1), (0, 1), (-1, 0), (0, -1), (-1, 0), (1, 0), (1, 0), (-1, 0), (0, -1), (0, 1), (1, 0), (-1, 0), (1, 0), (0, -1), (0, 1), (0, -1), (-1, 0), (0, 1), (-1, 0), (1, 0), (0, -1), (-1, 0), (0, 1), (1, 0), (0, -1), (1, 0), (0, 1), (0, 1), (0, -1), (-1, 0), (1, 0), (-1, 0), (-1, 0), (1, 0), (0, 1), (1, 0), (-1, 0), (-1, 0), (1, 0), (-1, 0), (1, 0), (0, -1), (0, 1), (0, -1), (0, -1), (0, 1), (-1, 0), (0, 1), (0, -1), (1, 0), (1, 0), (-1, 0), (0, 1), (0, -1), (0, 1), (1, 0), (0, -1), (0, 1), (0, -1), (0, -1), (0, 1), (0, 1), (-1, 0), (1, 0), (-1, 0), (0, -1), (1, 0), (-1, 0), (-1, 0), (0, -1), (1, 0), (0, 1), (-1, 0), (1, 0), (1, 0), (0, -1), (-1, 0), (0, 1), (1, 0), (-1, 0), (1, 0), (-1, 0), (0, -1), (1, 0), (0, 1), (0, -1), (-1, 0), (-1, 0), (1, 0), (0, 1), (0, -1), (0, 1), (-1, 0), (1, 0), (1, 0), (0, -1), (0, 1), (-1, 0), (0, -1), (1, 0), (-1, 0), (0, 1), (0, 1), (1, 0), (-1, 0), (0, -1), (0, 1), (-1, 0), (1, 0), (0, -1), (1, 0), (-1, 0), (-1, 0), (1, 0), (0, 1), (1, 0), (0, -1), (-1, 0), (-1, 0), (1, 0), (0, -1), (0, 1), (0, 1), (-1, 0), (0, -1), (0, -1), (0, 1), (0, 1), (0, -1), (0, 1), (1, 0), (1, 0), (-1, 0), (0, -1), (-1, 0), (0, -1), (1, 0), (0, 1), (1, 0), (-1, 0), (0, -1), (-1, 0), (0, 1), (1, 0), (0, -1), (1, 0), (0, 1), (0, 1), (0, -1), (0, -1), (0, 1), (-1, 0), (-1, 0), (1, 0), (0, 1), (0, -1), (1, 0), (-1, 0), (0, -1), (0, 1), (0, -1), (-1, 0), (0, 1), (0, -1), (1, 0), (1, 0), (0, 1), (-1, 0), (0, -1), (1, 0), (0, 1), (0, 1), (-1, 0), (-1, 0), (0, -1), (1, 0), (-1, 0), (1, 0), (1, 0), (0, 1), (0, -1), (-1, 0), (0, 1), (0, -1), (0, -1), (1, 0), (0, 1), (-1, 0), (1, 0), (-1, 0), (1, 0), (0, 1), (-1, 0), (1, 0), (-1, 0), (1, 0), (-1, 0), (-1, 0), (1, 0), (1, 0), (0, -1), (-1, 0), (-1, 0), (0, -1), (1, 0), (0, 1), (0, 1), (-1, 0), (0, -1), (0, -1), (1, 0), (0, 1), (-1, 0), (1, 0), (0, 1), (1, 0), (-1, 0), (-1, 0), (1, 0), (-1, 0), (1, 0), (1, 0), (-1, 0), (0, -1), (0, -1), (0, 1), (0, 1), (0, -1), (1, 0), (-1, 0), (-1, 0), (0, 1), (0, -1), (1, 0), (0, 1), (1, 0), (-1, 0), (-1, 0), (0, -1), (0, -1), (1, 0), (0, 1), (0, -1), (-1, 0), (1, 0), (1, 0), (0, 1), (-1, 0), (0, -1), (1, 0), (-1, 0), (1, 0), (0, 1), (0, -1), (-1, 0), (0, 1), (0, 1), (0, -1), (1, 0), (0, 1), (-1, 0), (-1, 0), (1, 0), (0, -1), (0, 1), (1, 0), (-1, 0), (-1, 0), (0, -1), (1, 0), (1, 0), (0, -1), (0, 1), (0, 1)
 
--[[User:Alec|Alec]] 14:42, 9 January 2010 (UTC)

Revision as of 11:09, 9 January 2010

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