Ostrowski numeration

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In mathematics, Ostrowski numeration, named after

.

Fix a positive irrational number α with continued fraction expansion [a₀; a₁, a₂, ...]. Let (q_n) be the sequence of denominators of the convergents p_n/q_n to α: so q_n = a_nq_n−1 + q_n−2. Let α_n denote Tⁿ(α) where T is the Gauss map T(x) = {1/x}, and write β_n = (−1)ⁿ⁺¹ α₀ α₁ ... α_n: we have β_n = a_nβ_n−1 + β_n−2.

Real number representations

Every positive real x can be written as

${\displaystyle x=\sum _{n=1}^{\infty }b_{n}\beta _{n}\ }$

where the integer coefficients 0 ≤ b_n ≤ a_n and if b_n = a_n then b_n−1 = 0.

Integer representations

Every positive integer N can be written uniquely as

${\displaystyle N=\sum _{n=1}^{k}b_{n}q_{n}\ }$

where the integer coefficients 0 ≤ b_n ≤ a_n and if b_n = a_n then b_n−1 = 0.

If α is the

of positive integers as a sum of distinct non-consecutive Fibonacci numbers.

See also

• Complete sequence

References

• Allouche, Jean-Paul;
  Zbl 1086.11015
  ..
• Epifanio, C.; Frougny, C.; Gabriele, A.; Mignosi, F.;
  Zbl 1237.68134
  .
• JFM 48.0197.04
  .
• Pytheas Fogg, N. (2002).
  Zbl 1014.11015
  .