In mathematical analysis, the Foias constant is a real number named after Ciprian Foias.
It is defined in the following way: for every real number x<sub>1</sub> > 0, there is a sequence defined by the recurrence relation
for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x<sub>1</sub> = α then the sequence diverges to infinity. For all other values of x<sub>1</sub>, the sequence is divergent as well, but it has two accumulation points: 1 and infinity. Numerically, it is
No closed form for the constant is known.
When x<sub>1</sub> = α then the growth rate of the sequence (x<sub>n</sub>) is given by the limit
where "log" denotes the natural logarithm.
The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences.