T2(x) = -x: Difference between revisions
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Sequence of length 614. |
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This sequence, of length <math> | This sequence, of length <math>614</math>, satisfies <math>x_{2n} = -x_n</math> for all <math>n</math>: | ||
0+--+-++--+--+-++-++--+-++-++--+-++--+--+ | |||
-+--++--+-++--+-++++---+--+-++-++--+-++-- | |||
+--++-+--+-++-+--++--+-++-++--++-+-+---++ | |||
-+-+--++-++--++-+-++--+--+-++--+--+-++-++ | |||
--+++----+++--+-+--++--+-++--++++--+--+-+ | |||
+-++--+-++--+--+-++---++--++--++-+-++-+-- | |||
++--++-++-+----+++-+-++-+--+--++--++---++ | |||
-++--+-++--+--+-++-++--+++---++-+-++--+-- | |||
+++--+-+--+++-+-+--+-+--+++--++---+-++--- | |||
++++--++--+--+-++-+-+-+--+-++--+++--++--+ | |||
-++--+--+++---+--+-++++---+-++-++----++-+ | |||
++----++-++-+-++-+--+--++--+--+-+++--+++- | |||
-+--+++----+++---++--++-+++-+--+---++--++ | |||
+-+-++---+++---+-++-+-++-+----++-+++--+-- | |||
++---++-++-++----++-++--+++---+--+-++--+- | |||
Here's one of length <math>594</math>, the previous record: | |||
0+--+-++--+--+-++-++--+-++--+--+-++ | 0+--+-++--+--+-++-++--+-++--+--+-++ | ||
| Line 19: | Line 37: | ||
-++++-+-+--+--++---++++--++--+-++-+ | -++++-+-+--+--++---++++--++--+-++-+ | ||
Here's one of length <math>584</math>, | Here's one of length <math>584</math>, the record before that: | ||
0+--+-++--++---++-+--+-++++-+--+-++---+-++--+ | 0+--+-++--++---++-+--+-++++-+--+-++---+-++--+ | ||
| Line 34: | Line 52: | ||
++--+--+-+--++-++--++--++-+-++---+++--+--++-+ | ++--+--+-+--++-++--++--++-+-++---+++--+--++-+ | ||
+---+++--+-+--+-+-++-+-+-++-+-+--+-+--+-++--- | +---+++--+-+--+-+-++-+-+-++-+-+--+-+--+-++--- | ||
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. | |||
Latest revision as of 12:38, 10 January 2010
This sequence, of length [math]\displaystyle{ 614 }[/math], satisfies [math]\displaystyle{ x_{2n} = -x_n }[/math] for all [math]\displaystyle{ n }[/math]:
0+--+-++--+--+-++-++--+-++-++--+-++--+--+ -+--++--+-++--+-++++---+--+-++-++--+-++-- +--++-+--+-++-+--++--+-++-++--++-+-+---++ -+-+--++-++--++-+-++--+--+-++--+--+-++-++ --+++----+++--+-+--++--+-++--++++--+--+-+ +-++--+-++--+--+-++---++--++--++-+-++-+-- ++--++-++-+----+++-+-++-+--+--++--++---++ -++--+-++--+--+-++-++--+++---++-+-++--+-- +++--+-+--+++-+-+--+-+--+++--++---+-++--- ++++--++--+--+-++-+-+-+--+-++--+++--++--+ -++--+--+++---+--+-++++---+-++-++----++-+ ++----++-++-+-++-+--+--++--+--+-+++--+++- -+--+++----+++---++--++-+++-+--+---++--++ +-+-++---+++---+-++-+-++-+----++-+++--+-- ++---++-++-++----++-++--+++---+--+-++--+-
Here's one of length [math]\displaystyle{ 594 }[/math], the previous record:
0+--+-++--+--+-++-++--+-++--+--+-++ --+-++-+--++--+-++-++--+-++--+--+-+ +-++--+--+-++--++++--+--+-++-++---- +++-+--+-++--+--+-++-++--+-++-++--+ -++--+--+-++--+-++-++--+--+-++-+-+- +--+++---+-++-++--+--+--+-++-+++-+- ---+-+++-++-+---+--+-++-++--+++---+ --+-+++-+--+-++-++--+-++--+--+-++-+ +----++-++--+-++--+--+-++-++--++-+- ++--+--+-++-++-+---+--+-++--++--++- -++-++---+-++-+-++-++--+-++--+--+-+ +--+-++--+-++-++--+--+--+-+-++--++- +-+-++--++-+-+--+--+-++--++++-+---+ -++-++--+-++--+--+++--+---++-+++--+ +-+---++-+-+--++--+++---++-+-++--+- -+-++--++---++-++--+-+--+++-+-++--- -++++-+-+--+--++---++++--++--+-++-+
Here's one of length [math]\displaystyle{ 584 }[/math], the record before that:
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.
