Ostrowski numeration

<h2 id="real-number-representations">Real number representations</h2>
Every positive real x can be written as

x
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          b
          
            n
          
        
        
          β
          
            n
          
        
         
      
    
    {\displaystyle x=\sum _{n=1}^{\infty }b_{n}\beta _{n}\ }

where the integer coefficients 0 ≤ bn ≤ an and if bn = an then bn−1 = 0.

<h2 id="integer-representations">Integer representations</h2>
Every positive integer N can be written uniquely as

N
        =
        
          ∑
          
            n
            =
            1
          
          
            k
          
        
        
          b
          
            n
          
        
        
          q
          
            n
          
        
         
      
    
    {\displaystyle N=\sum _{n=1}^{k}b_{n}q_{n}\ }

where the integer coefficients 0 ≤ bn ≤ an and if bn = an then bn−1 = 0.
If α is the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a>, then all the partial quotients an are equal to 1, the denominators qn are the <a href="/facts/Fibonacci_numbers/mqxpAUQD">Fibonacci numbers</a> and we recover <a href="/facts/Zeckendorf%27s_theorem/hg8s2ycr">Zeckendorf's theorem</a> on the <a href="/facts/Fibonacci_representation/GXCLAf7t">Fibonacci representation</a> of positive integers as a sum of distinct non-consecutive Fibonacci numbers.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Complete_sequence/k2PaxE9Y">Complete sequence</a></li></ul>

<ul><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. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1086.11015">1086.11015</a>..</li>
<li>Epifanio, C.; Frougny, C.; Gabriele, A.; Mignosi, F.; <a href="/facts/Jeffrey_Shallit/4bmJgyIH">Shallit, J.</a> (2012). <a href="https://doi.org/10.1016%2Fj.dam.2011.10.029">"Sturmian graphs and integer representations over numeration systems"</a>. Discrete Appl. Math. 160 (4–5): 536–547. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.dam.2011.10.029">10.1016/j.dam.2011.10.029</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0166-218X">0166-218X</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1237.68134">1237.68134</a>.</li>
<li><a href="/facts/Alexander_Ostrowski/r45sj25p">Ostrowski, Alexander</a> (1921). "Bemerkungen zur Theorie der diophantischen Approximationen". Hamb. Abh. (in German). 1: 77–98. <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:48.0197.04">48.0197.04</a>.</li>
<li>Pytheas Fogg, N. (2002). <a href="/facts/Val%C3%A9rie_Berth%C3%A9/0puS3RSe">Berthé, Valérie</a>; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-44141-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1014.11015">1014.11015</a>.</li></ul>

Ostrowski numeration open-in-new

Ostrowski numeration