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:
“Don: Why are they closed? They’re all closed, every one of them.
Pawnbroker: Sure they are. It’s Yom Kippur.
Don: It’s what?
Pawnbroker: It’s Yom Kippur, a Jewish holiday.
Don: It is? So what about Kelly’s and Gallagher’s?
Pawnbroker: They’re closed, too. We’ve got an agreement. They keep closed on Yom Kippur and we don’t open on St. Patrick’s.”
—Billy Wilder (b. 1906)
“Whoever denies that he possesses vanity generally possesses it in so brutal a form that he instinctively shuts his eyes in its presence, so as not to have to look down upon himself.”
—Friedrich Nietzsche (1844–1900)
“The formula for achieving a successful relationship is simple: you should treat all disasters as if they were trivialities but never treat a triviality as if it were a disaster.”
—Quentin Crisp (b. 1908)