Sylvester's Sequence - Closed Form Formula and Asymptotics

Closed Form Formula and Asymptotics

The Sylvester numbers grow doubly exponentially as a function of n. Specifically, it can be shown that

for a number E that is approximately 1.264084735305302. This formula has the effect of the following algorithm:

s0 is the nearest integer to E2; s1 is the nearest integer to E4; s2 is the nearest integer to E8; for sn, take E2, square it n more times, and take the nearest integer.

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

The double-exponential growth of the Sylvester sequence is unsurprising if one compares it to the sequence of Fermat numbers Fn; the Fermat numbers are usually defined by a doubly exponential formula, but they can also be defined by a product formula very similar to that defining Sylvester's sequence:

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