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In [[mathematics]] the '''Baum–Sweet sequence''' is an infinite [[automatic sequence]] of 0s and 1s defined by the rule:


:''b''<sub>''n''</sub> = 1 if the binary representation of ''n'' contains no block of consecutive 0s of odd length;
:''b''<sub>''n''</sub> = 0 otherwise;


for ''n'' &ge; 0.<ref>{{MathWorld|urlname=Baum-SweetSequence|title=Baum–Sweet Sequence}}</ref>
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For example, ''b''<sub>4</sub> = 1 because the binary representation of 4 is 100, which only contains one block of consecutive 0s of length 2; whereas ''b''<sub>5</sub> = 0 because the binary representation of 5 is 101, which contains a block of consecutive 0s of length 1.
 
Starting at ''n'' = 0, the first few terms of the Baum–Sweet sequence are:
 
:1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1 ... {{OEIS|A086747}}
 
The properties of the sequence were first studied by L.E. Baum and M.M. Sweet in 1976.<ref>Baum, L. E. and Sweet, M. M. ''Continued Fractions of Algebraic Power Series in Characteristic 2'' Ann. Math. 103, 593-610, 1976.</ref>
 
==Properties==
The Baum–Sweet sequence can be generated by a three state [[Finite state machine|automaton]].<ref>[http://www.emis.de/journals/SLC/opapers/s30allouche.pdf Finite automata and arithmetic],
Jean-Paul Allouche</ref>
 
The value of term ''b''<sub>''n''</sub> in the Baum–Sweet sequence can be found recursively as follows. If ''n'' = ''m''&middot;4<sup>''k''</sup>, where ''m'' is not divisible by 4, then
 
:<math>b_n =
\begin{cases}
0 & \text{if } m \text{ is even} \\
b_{(m-1)/2} & \text{if } m \text{ is odd}.
\end{cases}</math>
 
Thus b<sub>76</sub> = ''b''<sub>9</sub> = ''b''<sub>4</sub> = ''b''<sub>0</sub> = 1, which can be verified by observing that the binary representation of 76, which is 1001100, contains no consecutive blocks of 0s with odd length.
 
The Baum–Sweet word 1101100101001001..., which is created by concatenating the terms of the Baum–Sweet sequence, is a fixed point of the morphism or [[string substitution]] rules
 
:00 '''→''' 0000
:01 '''→''' 1001
:10 '''→''' 0100
:11 '''→''' 1101
 
as follows:
 
:11 '''→''' 1101 '''→''' 11011001 '''→''' 1101100101001001 '''→''' 11011001010010011001000001001001 ...
 
From the morphism rules it can be seen that the Baum–Sweet word contains blocks of consecutive 0s of any length (''b''<sub>''n''</sub>&nbsp;=&nbsp;0 for all 2<sup>''k''</sup> integers in the range 5.2<sup>''k''</sup> &le; ''n''&nbsp;<&nbsp;6.2<sup>''k''</sup>), but it contains no block of three consecutive 1s.
 
The Baum–Sweet sequence is the sequence of coefficients of the unique solution of the cubic equation ''f''<sup>&nbsp;3</sup>&nbsp;+&nbsp;''Xf''&nbsp;+&nbsp;1&nbsp;=&nbsp;0 in the field '''F'''<sub>2</sub>((''X''<sup>&nbsp;&minus;1</sup>)) of formal [[Laurent series]] over '''F'''<sub>2</sub>.<ref>Graham Everest ''Recurrence Sequences'' AMS 2003, p&nbsp;236</ref>
 
==Notes==
{{reflist}}
{{refbegin}}
{{refend}}
 
==References==
*{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
 
{{DEFAULTSORT:Baum-Sweet sequence}}
[[Category:Binary sequences]]

Latest revision as of 14:15, 19 September 2014


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