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.Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Calude and G. Păun). London: Springer-Verlag, pp. 119–126, 2000. Numerically, it is . The constant can be computed by solving backwards:

This recursion converges for any starting value .

When x1 = α then the growth rate of the sequence (xn) 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.

See also Mathematical constant

Notes and references