T2(x) = -x: Difference between revisions
m T2(x) = -T(x) moved to T2(x) = -x |
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A more colourful display of this sequence can be found [http://thomas1111.wordpress.com/2010/01/10/a-sequence-of-584-elements-with-t2x-tx/ on this page]. | A more colourful display of this sequence can be found [http://thomas1111.wordpress.com/2010/01/10/a-sequence-of-584-elements-with-t2x-tx/ on this page]. | ||
This sequence has various multiplicative properties that are not implied by the basic constraint. Writing <math>a=\pm b</math> to mean that the beginning of the a-sequence closely resembles the beginning of the b-sequence, we have the following relationships: 2=5=17, 3=-15, 7=-9, 11=-13. In addition, the 3-sequence starts pretty similarly to the 13-sequence, and consequently the 11-sequence starts pretty similarly to the 15-sequence. | |||
Revision as of 11:20, 10 January 2010
This sequence, of length [math]\displaystyle{ 594 }[/math], satisfies [math]\displaystyle{ x_{2n} = -x_n }[/math] for all [math]\displaystyle{ n }[/math]:
0+--+-++--+--+-++-++--+-++--+--+-++ --+-++-+--++--+-++-++--+-++--+--+-+ +-++--+--+-++--++++--+--+-++-++---- +++-+--+-++--+--+-++-++--+-++-++--+ -++--+--+-++--+-++-++--+--+-++-+-+- +--+++---+-++-++--+--+--+-++-+++-+- ---+-+++-++-+---+--+-++-++--+++---+ --+-+++-+--+-++-++--+-++--+--+-++-+ +----++-++--+-++--+--+-++-++--++-+- ++--+--+-++-++-+---+--+-++--++--++- -++-++---+-++-+-++-++--+-++--+--+-+ +--+-++--+-++-++--+--+--+-+-++--++- +-+-++--++-+-+--+--+-++--++++-+---+ -++-++--+-++--+--+++--+---++-+++--+ +-+---++-+-+--++--+++---++-+-++--+- -+-++--++---++-++--+-+--+++-+-++--- -++++-+-+--+--++---++++--++--+-++-+
Here's one of length [math]\displaystyle{ 584 }[/math], which held the record for about ten minutes:
0+--+-++--++---++-+--+-++++-+--+-++---+-++--+ +-+-+-+-++---+-++--++---++-+++---++-+--+++--- -++--+++-++---+--+-++++-+---+--+-++-++---++-+ ++--+--+--+-+-++-+++----+++--+-++-+-+--+++-+- +----++-+--+-+-+++-+-++-+-+---+-+--+++-+-+-+- -++--+++-+---+++--++--+-++--+--+-+-++-+--++-+ -+--+++---+++---+++--++--++----+++-+---++-+-+ ++--+---++-+--+++-+--++--++--+++--+-+--++--++ -++-+-+-+--+--++--+++--++---++---+-+++-++--+- -++--++--+-+++--++--++-+--+-+-++---++-++---++ +----++-++-+-+-++--++-+-+--+-+--+-+--+-++-++- ++--+--+-+--++-++--++--++-+-++---+++--+--++-+ +---+++--+-+--+-+-++-+-+-++-+-+--+-+--+-++---
A more colourful display of this sequence can be found on this page.
This sequence has various multiplicative properties that are not implied by the basic constraint. Writing [math]\displaystyle{ a=\pm b }[/math] to mean that the beginning of the a-sequence closely resembles the beginning of the b-sequence, we have the following relationships: 2=5=17, 3=-15, 7=-9, 11=-13. In addition, the 3-sequence starts pretty similarly to the 13-sequence, and consequently the 11-sequence starts pretty similarly to the 15-sequence.
