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:
Read more about this topic: Sylvester's Sequence
Famous quotes containing the words closed, form and/or formula:
“Thus piteously Love closed what he begat:
The union of this ever-diverse pair!
These two were rapid falcons in a snare,
Condemned to do the flitting of the bat.”
—George Meredith (1828–1909)
“To do the opposite of something is also a form of imitation, namely an imitation of its opposite.”
—G.C. (Georg Christoph)
“In the most desirable conditions, the child learns to manage anxiety by being exposed to just the right amounts of it, not much more and not much less. This optimal amount of anxiety varies with the child’s age and temperament. It may also vary with cultural values.... There is no mathematical formula for calculating exact amounts of optimal anxiety. This is why child rearing is an art and not a science.”
—Alicia F. Lieberman (20th century)