Sylvester's sequence: is there an exact closed form?

Question

I'm afraid this is one of those "amateur mathematician with no journal access" questions. Anyhow, Wikipedia (here) and OEIS give this closed form for the terms Sylvester's sequence:

$S_n = \lfloor E^{2^{n+1}} + \frac12\rfloor$ (where $E \approx 1.264$).

Is this the best we can do for a closed form, or is there an exact formula known? Is it known that there is no such formula?

Answer 1

To quote that Wikipedia article (emphasis mine):

This would only be a practical algorithm if we had a better way of calculating $E$ to the requisite number of places than calculating $s_n$ and taking its repeated square root.