  We extend the classical Ostrowski numeration systems, closely related to
Sturmian words, by allowing a wider range of coefficients, so that possible
representations of a number $n$ better reflect the structure of the associated
Sturmian word. In particular, this extended numeration system helps to catch
occurrences of palindromes in a characteristic Sturmian word and thus to prove
for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni
and the author: If a word is not periodic, then for every $Q>0$ it has a prefix
which cannot be decomposed to a concatenation of at most $Q$ palindromes.
