Fibonacci word

<h2 id="definition">Definition</h2>
Let 
 
 
 
 
 S
 
 0
 
 
 
 
 {\displaystyle S_{0}}
 
 be "0" and 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S_{1}}
 
 be "01". Now 
 
 
 
 
 S
 
 n
 
 
 =
 
 S
 
 n
 −
 1
 
 
 
 S
 
 n
 −
 2
 
 
 
 
 {\displaystyle S_{n}=S_{n-1}S_{n-2}}
 
 (the concatenation of the previous sequence and the one before that).
The infinite Fibonacci word is the limit 
 
 
 
 
 S
 
 ∞
 
 
 
 
 {\displaystyle S_{\infty }}
 
, that is, the (unique) infinite sequence that contains each 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n}}
 
, for finite 
 
 
 
 n
 
 
 {\displaystyle n}
 
, as a prefix.
Enumerating items from the above definition produces:

S
          
            0
          
        
      
    
    {\displaystyle S_{0}}
  
    0

S
          
            1
          
        
      
    
    {\displaystyle S_{1}}
  
    01

S
          
            2
          
        
      
    
    {\displaystyle S_{2}}
  
    010

S
          
            3
          
        
      
    
    {\displaystyle S_{3}}
  
    01001

S
          
            4
          
        
      
    
    {\displaystyle S_{4}}
  
    01001010

S
 
 5
 
 
 
 
 {\displaystyle S_{5}}
 
    0100101001001
...
The first few elements of the infinite Fibonacci word are:
0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, ... (sequence A003849 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>)

<h2 id="closed-form-expression-for-individual-digits">Closed-form expression for individual digits</h2>
The nth digit of the word is 
 
 
 
 2
 +
 ⌊
 n
 φ
 ⌋
 −
 ⌊
 (
 n
 +
 1
 )
 φ
 ⌋
 
 
 {\displaystyle 2+\lfloor n\varphi \rfloor -\lfloor (n+1)\varphi \rfloor }
 
 where 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 is the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a> and 
 
 
 
 ⌊
 
  
 ⌋
 
 
 {\displaystyle \lfloor \,\ \rfloor }
 
 is the <a href="/facts/Floor_function/ssehyMMf">floor function</a> (sequence A003849 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope 
 
 
 
 1
 
 /
 
 φ
 
 
 {\displaystyle 1/\varphi }
 
 or 
 
 
 
 φ
 −
 1
 
 
 {\displaystyle \varphi -1}
 
. See the figure above.

<h2 id="substitution-rules">Substitution rules</h2>
Another way of going from Sn to Sn +1 is to replace each symbol 0 in Sn with the pair of consecutive symbols 0, 1 in Sn +1, and to replace each symbol 1 in Sn with the single symbol 0 in Sn +1.
Alternatively, one can imagine directly generating the entire infinite Fibonacci word by the following process: start with a cursor pointing to the single digit 0. Then, at each step, if the cursor is pointing to a 0, append 1, 0 to the end of the word, and if the cursor is pointing to a 1, append 0 to the end of the word. In either case, complete the step by moving the cursor one position to the right.
A similar infinite word, sometimes called the rabbit sequence, is generated by a similar infinite process with a different replacement rule: whenever the cursor is pointing to a 0, append 1, and whenever the cursor is pointing to a 1, append 0, 1. The resulting sequence begins

0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, ...
However this sequence differs from the Fibonacci word only trivially, by swapping 0s for 1s and shifting the positions by one.
A closed form expression for the so-called rabbit sequence:
The nth digit of the word is 
 
 
 
 ⌊
 n
 φ
 ⌋
 −
 ⌊
 (
 n
 −
 1
 )
 φ
 ⌋
 −
 1.
 
 
 {\displaystyle \lfloor n\varphi \rfloor -\lfloor (n-1)\varphi \rfloor -1.}

<h2 id="discussion">Discussion</h2>
The word is related to the famous sequence of the same name (the <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci sequence</a>) in the sense that addition of integers in the <a href="/facts/Inductive_definition/9RfhFw0R">inductive definition</a> is replaced with string concatenation. This causes the length of Sn to be Fn +2, the (n +2)nd Fibonacci number. Also the number of 1s in Sn is Fn and the number of 0s in Sn is Fn +1.

<h2 id="other-properties">Other properties</h2>
<ul><li>The infinite Fibonacci word is not periodic and not ultimately periodic.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>The last two letters of a Fibonacci word are alternately "01" and "10".</li>
<li>Suppressing the last two letters of a Fibonacci word, or prefixing the complement of the last two letters, creates a <a href="/facts/Palindrome/PHo33zqN">palindrome</a>. Example: 01S4 = 0101001010 is a palindrome. The <a href="/facts/Palindromic_density/PHo33zqN">palindromic density</a> of the infinite Fibonacci word is thus 1/φ, where φ is the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a>: this is the largest possible value for aperiodic words.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>The infinite Fibonacci word is a <a href="/facts/Balanced_sequence/RHKkS4x6">balanced sequence</a>: Take two <a href="/facts/Substring/dM5Rqg6X">factors</a> of the same length anywhere in the Fibonacci word. The difference between their <a href="/facts/Hamming_weight/dJIW7qg4">Hamming weights</a> (the number of occurrences of "1") never exceeds 1.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>The subwords 11 and 000 never occur.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>The <a href="/facts/Complexity_function/RHKkS4x6">complexity function</a> of the infinite Fibonacci word is n + 1: it contains n + 1 distinct subwords of length n. Example: There are 4 distinct subwords of length 3: "001", "010", "100" and "101". Being also non-periodic, it is then of "minimal complexity", and hence a <a href="/facts/Sturmian_word/tbwHJPb6">Sturmian word</a>,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> with slope 
 
 
 
 1
 
 /
 
 φ
 
 
 {\displaystyle 1/\varphi }
 
. The infinite Fibonacci word is the <a href="/facts/Sturmian_word/tbwHJPb6">standard word</a> generated by the <a href="/facts/Sturmian_word/tbwHJPb6">directive sequence</a> (1,1,1,....).</li>
<li>The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often.</li>
<li>If 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a subword of the infinite Fibonacci word, then so is its reversal, denoted 
 
 
 
 
 u
 
 R
 
 
 
 
 {\displaystyle u^{R}}
 
.</li>
<li>If 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a subword of the infinite Fibonacci word, then the least period of 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a Fibonacci number.</li>
<li>The concatenation of two successive Fibonacci words is "almost commutative". 
 
 
 
 
 S
 
 n
 +
 1
 
 
 =
 
 S
 
 n
 
 
 
 S
 
 n
 −
 1
 
 
 
 
 {\displaystyle S_{n+1}=S_{n}S_{n-1}}
 
 and 
 
 
 
 
 S
 
 n
 −
 1
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n-1}S_{n}}
 
 differ only by their last two letters.</li>
<li>The number 0.010010100..., whose digits are built with the digits of the infinite Fibonacci word, is <a href="/facts/Transcendental_number/IrB7JppM">transcendental</a>.</li>
<li>The letters "1" can be found at the positions given by the successive values of the Upper Wythoff sequence (sequence A001950 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>): 
 
 
 
 ⌊
 n
 
 φ
 
 2
 
 
 ⌋
 
 
 {\displaystyle \lfloor n\varphi ^{2}\rfloor }
 
</li>
<li>The letters "0" can be found at the positions given by the successive values of the Lower Wythoff sequence (sequence A000201 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>): 
 
 
 
 ⌊
 n
 φ
 ⌋
 
 
 {\displaystyle \lfloor n\varphi \rfloor }
 
</li>
<li>The distribution of 
 
 
 
 n
 =
 
 F
 
 k
 
 
 
 
 {\displaystyle n=F_{k}}
 
 points on the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>, placed consecutively clockwise by the golden angle 
 
 
 
 
 
 
 2
 π
 
 
 φ
 
 2
 
 
 
 
 
 
 {\displaystyle {\frac {2\pi }{\varphi ^{2}}}}
 
, generates a pattern of two lengths 
 
 
 
 
 
 
 2
 π
 
 
 φ
 
 k
 −
 1
 
 
 
 
 ,
 
 
 
 2
 π
 
 
 φ
 
 k
 
 
 
 
 
 
 {\displaystyle {\frac {2\pi }{\varphi ^{k-1}}},{\frac {2\pi }{\varphi ^{k}}}}
 
 on the unit circle. Although the above generating process of the Fibonacci word does not correspond directly to the successive division of circle segments, this pattern is 
 
 
 
 
 S
 
 k
 −
 1
 
 
 
 
 {\displaystyle S_{k-1}}
 
 if the pattern starts at the point nearest to the first point in clockwise direction, whereupon 0 corresponds to the long distance and 1 to the short distance.</li>
<li>The infinite Fibonacci word contains repetitions of 3 successive identical subwords, but none of 4.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The <a href="/facts/Critical_exponent_of_a_word/TcoWWXA9">critical exponent</a> for the infinite Fibonacci word is 
 
 
 
 2
 +
 φ
 ≈
 3.618
 
 
 {\displaystyle 2+\varphi \approx 3.618}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> It is the smallest index (or critical exponent) among all Sturmian words.</li>
<li>The infinite Fibonacci word is often cited as the <a href="/facts/Worst_case/Ed9ekH30">worst case</a> for algorithms detecting repetitions in a string.</li>
<li>The infinite Fibonacci word is a <a href="/facts/Morphic_word/MgvrmJYM">morphic word</a>, generated in {0,1}* by the endomorphism 0 → 01, 1 → 0.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>The nth element of a Fibonacci word, 
 
 
 
 
 s
 
 n
 
 
 
 
 {\displaystyle s_{n}}
 
, is 1 if the <a href="/facts/Zeckendorf%27s_theorem/hg8s2ycr">Zeckendorf representation</a> (the sum of a specific set of Fibonacci numbers) of n includes a 1, and 0 if it does not include a 1.</li>
<li>The digits of the Fibonacci word may be obtained by taking the sequence of <a href="/facts/Fibbinary_number/tS3lJfTp">fibbinary numbers</a> <a href="/facts/Modulo_operation/V5xPUluD">modulo</a> 2.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li></ul>
<h2 id="applications">Applications</h2>
Fibonacci based constructions are currently used to model physical systems with aperiodic order such as <a href="/facts/Quasicrystal/evHcQeCp">quasicrystals</a>, and in this context the Fibonacci word is also called the Fibonacci quasicrystal.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Crystal growth techniques have been used to grow Fibonacci layered crystals and study their light scattering properties.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Mathematics_and_art/kT01G7la">Mathematics and art</a></li>
<li><a href="/facts/Tribonacci_word/oVKETtAY">Tribonacci word</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Adamczewski, Boris; Bugeaud, Yann (2010), "8. Transcendence and diophantine approximation", in <a href="/facts/Val%C3%A9rie_Berth%C3%A9/0puS3RSe">Berthé, Valérie</a>; Rigo, Michael (eds.), Combinatorics, automata, and number theory, Encyclopedia of Mathematics and its Applications, vol. 135, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, p. 443, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-51597-9, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1271.11073">1271.11073</a>.</li>
<li>Allouche, Jean-Paul; <a href="/facts/Jeffrey_Shallit/4bmJgyIH">Shallit, Jeffrey</a> (2003), Automatic Sequences: Theory, Applications, Generalizations, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-82332-6.</li>
<li>Berstel, Jean (1986), <a href="https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf">"Fibonacci Words – A Survey"</a> (PDF), in Rozenberg, G.; Salomaa, A. (eds.), The Book of L, Springer, pp. 13–27, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-95486-3_2">10.1007/978-3-642-95486-3_2</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783642954863</li>
<li><a href="/facts/Enrico_Bombieri/MU6xU5vr">Bombieri, E.</a>; <a href="/facts/Jean_Taylor/05XPoDt5">Taylor, J. E.</a> (1986), <a href="https://hal.archives-ouvertes.fr/jpa-00225713/file/ajp-jphyscol198647C303.pdf">"Which distributions of matter diffract? An initial investigation"</a> (PDF), Le Journal de Physique, 47 (C3): 19–28, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1051%2Fjphyscol%3A1986303">10.1051/jphyscol:1986303</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0866320">0866320</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:54194304">54194304</a>.</li>
<li>Dharma-wardana, M. W. C.; MacDonald, A. H.; Lockwood, D. J.; Baribeau, J.-M.; Houghton, D. C. (1987), "Raman scattering in Fibonacci superlattices", <a href="/facts/Physical_Review_Letters/rNP3XvYq">Physical Review Letters</a>, 58 (17): 1761–1765, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1987PhRvL..58.1761D">1987PhRvL..58.1761D</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2Fphysrevlett.58.1761">10.1103/physrevlett.58.1761</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/10034529">10034529</a>.</li>
<li><a href="/facts/Clark_Kimberling/WMRRw662">Kimberling, Clark</a> (2004), "Ordering words and sets of numbers: the Fibonacci case", in Howard, Frederic T. (ed.), Applications of Fibonacci Numbers, Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, Dordrecht: Kluwer Academic Publishers, pp. 137–144, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-0-306-48517-6_14">10.1007/978-0-306-48517-6_14</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-90-481-6545-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2076798">2076798</a>.</li>
<li><a href="/facts/M._Lothaire/pC43ftpJ">Lothaire, M.</a> (1997), Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 17 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-59924-5.</li>
<li><a href="/facts/M._Lothaire/pC43ftpJ">Lothaire, M.</a> (2011), Algebraic Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 90, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-18071-9. Reprint of the 2002 hardback.</li>
<li>de Luca, Aldo (1995), "A division property of the Fibonacci word", <a href="/facts/Information_Processing_Letters/e2u7fEKe">Information Processing Letters</a>, 54 (6): 307–312, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0020-0190%2895%2900067-M">10.1016/0020-0190(95)00067-M</a>.</li>
<li>Mignosi, F.; Pirillo, G. (1992), <a href="https://scholar.archive.org/work/kzxlsp5mirgjhfojzatmrkogii">"Repetitions in the Fibonacci infinite word"</a>, Informatique Théorique et Application, 26 (3): 199–204, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1051%2Fita%2F1992260301991">10.1051/ita/1992260301991</a>.</li>
<li>Ramírez, José L.; Rubiano, Gustavo N.; De Castro, Rodrigo (2014), "A generalization of the Fibonacci word fractal and the Fibonacci snowflake", <a href="/facts/Theoretical_Computer_Science_(journal)/Apbc0PDL">Theoretical Computer Science</a>, 528: 40–56, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1212.1368">1212.1368</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.tcs.2014.02.003">10.1016/j.tcs.2014.02.003</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3175078">3175078</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:17193119">17193119</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20081218041832/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrab.html">A detailed and accessible description, on Ron Knott's site</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>, <a href="https://mathworld.wolfram.com/RabbitSequence.html">"Rabbit Sequence"</a>, <a href="/facts/MathWorld/yc1jodbq">MathWorld</a></li>
<li><a href="https://www.youtube.com/watch?v=ZDGGEQqSXew">Fibonacci Word (First 200,000 bits)</a> on <a href="/facts/YouTube_video_(identifier)/db276jst">YouTube</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Berstel (1986), p. 13. - Berstel, Jean (1986), "Fibonacci Words – A Survey" (PDF), in Rozenberg, G.; Salomaa, A. (eds.), The Book of L, Springer, pp. 13–27, doi:10.1007/978-3-642-95486-3_2, ISBN 9783642954863 <a href="https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf" target="_blank">https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Adamczewski & Bugeaud (2010). - Adamczewski, Boris; Bugeaud, Yann (2010), "8. Transcendence and diophantine approximation", in Berthé, Valérie; Rigo, Michael (eds.), Combinatorics, automata, and number theory, Encyclopedia of Mathematics and its Applications, vol. 135, Cambridge: Cambridge University Press, p. 443, ISBN 978-0-521-51597-9, Zbl 1271.11073 <a href="https://zbmath.org/?format=complete&q=an:1271.11073" target="_blank">https://zbmath.org/?format=complete&q=an:1271.11073</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Sloane, N. J. A. (ed.), "Sequence A003849", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lothaire (2011), p. 47. - Lothaire, M. (2011), Algebraic Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 90, Cambridge University Press, ISBN 978-0-521-18071-9 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">For the subwords that do occur, see Berstel (1986), pp. 14 and 18 (using the letters a and b in place of the digits 0 and 1) - Berstel, Jean (1986), "Fibonacci Words – A Survey" (PDF), in Rozenberg, G.; Salomaa, A. (eds.), The Book of L, Springer, pp. 13–27, doi:10.1007/978-3-642-95486-3_2, ISBN 9783642954863 <a href="https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf" target="_blank">https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">de Luca (1995). - de Luca, Aldo (1995), "A division property of the Fibonacci word", Information Processing Letters, 54 (6): 307–312, doi:10.1016/0020-0190(95)00067-M <a href="https://doi.org/10.1016%2F0020-0190%2895%2900067-M" target="_blank">https://doi.org/10.1016%2F0020-0190%2895%2900067-M</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Berstel (1986), p. 13. - Berstel, Jean (1986), "Fibonacci Words – A Survey" (PDF), in Rozenberg, G.; Salomaa, A. (eds.), The Book of L, Springer, pp. 13–27, doi:10.1007/978-3-642-95486-3_2, ISBN 9783642954863 <a href="https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf" target="_blank">https://www-igm.univ-mlv.fr/~berstel/Articles/1985BookOfL.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Allouche & Shallit (2003), p. 37. - Allouche, Jean-Paul; Shallit, Jeffrey (2003), Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, ISBN 978-0-521-82332-6 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Lothaire (2011), p. 11. - Lothaire, M. (2011), Algebraic Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 90, Cambridge University Press, ISBN 978-0-521-18071-9 <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Kimberling (2004). - Kimberling, Clark (2004), "Ordering words and sets of numbers: the Fibonacci case", in Howard, Frederic T. (ed.), Applications of Fibonacci Numbers, Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, Dordrecht: Kluwer Academic Publishers, pp. 137–144, doi:10.1007/978-0-306-48517-6_14, ISBN 978-90-481-6545-2, MR 2076798 <a href="https://doi.org/10.1007%2F978-0-306-48517-6_14" target="_blank">https://doi.org/10.1007%2F978-0-306-48517-6_14</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Bombieri & Taylor (1986). - Bombieri, E.; Taylor, J. E. (1986), "Which distributions of matter diffract? An initial investigation" (PDF), Le Journal de Physique, 47 (C3): 19–28, doi:10.1051/jphyscol:1986303, MR 0866320, S2CID 54194304 <a href="https://hal.archives-ouvertes.fr/jpa-00225713/file/ajp-jphyscol198647C303.pdf" target="_blank">https://hal.archives-ouvertes.fr/jpa-00225713/file/ajp-jphyscol198647C303.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Dharma-wardana et al. (1987). - Dharma-wardana, M. W. C.; MacDonald, A. H.; Lockwood, D. J.; Baribeau, J.-M.; Houghton, D. C. (1987), "Raman scattering in Fibonacci superlattices", Physical Review Letters, 58 (17): 1761–1765, Bibcode:1987PhRvL..58.1761D, doi:10.1103/physrevlett.58.1761, PMID 10034529 <a href="https://ui.adsabs.harvard.edu/abs/1987PhRvL..58.1761D" target="_blank">https://ui.adsabs.harvard.edu/abs/1987PhRvL..58.1761D</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Fibonacci word open-in-new

Fibonacci word