Section: New Results
Stern-Brocot and Fibonacci sequences
Participant : José Grimm.
We constructed an explicit bijection , first in the framework of the Bourbaki project (see above), then in Ssreflect. Every positive rational number can uniquely be written as a quotient . This result was established by Dijkstra who stated it in an obfuscated way. It was shown years before by Stern. It is possible to compute without computing numerator and denominator separately, by considering the sequences of bits of from left to right or from right to left. Truncating the binary expansion of yields a sequence of approximations to (this was studied by Brocot, and the so-called Stern-Brocot tree is an alternative representation of rational numbers). We implemented the work of Dijkstra and Stern in Coq [17] .
We also studied how a number can be represented by a sequence of other numbers (for instance as a sum of distinct Fibonacci numbers, with or without constraints). The number of ways of writing as a sum of powers of two, each power of two being used at most twice, is . These results are presented in [17] .