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Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansionstarts with the largest odd denominator unit fraction at most $n/d$, adds thelargest odd denominator unit fraction so the sum is at most $n/d$, andcontinues as long as the sum is less than $n/d$. It is an open question whetherthis expansion always has finitely many terms. Given a fixed positive integer$n$, we find all reduced fractions with numerator $n$ whose odd greedyexpansion has length $2$. Given $m-1$ odd positive integers, we find allrational numbers whose odd greedy expansion has length $m$ and begins withthese numbers as denominators. Given $m-2$ compatible odd positive integers, wefind an infinite family of rational numbers whose odd greedy expansion haslength $m$ and begins with these numbers as denominators.

Rational numbers with odd greedy expansion of fixed length