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 x1 > 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 x1 = α then the sequence diverges to infinity. For all other values of x1, the sequence is divergent as well, but it has two accumulation points: 1 and infinity.[1] Numerically, it is
No closed form for the constant is known.
When x1 = α then the growth rate of the sequence (xn) is given by the limit
where "log" denotes the natural logarithm.[1]
The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences.[3]
Foias constant.