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:
“My old Father used to have a saying that “If you make a bad bargain, hug it the tighter”; and it occurs to me, that if the bargain you have just closed [marriage] can possibly be called a bad one, it is certainly the most pleasant one for applying that maxim to, which my fancy can, by any effort, picture.”
—Abraham Lincoln (1809–1865)
“Humility is often only the putting on of a submissiveness by which men hope to bring other people to submit to them; it is a more calculated sort of pride, which debases itself with a design of being exalted; and though this vice transform itself into a thousand several shapes, yet the disguise is never more effectual nor more capable of deceiving the world than when concealed under a form of humility.”
—François, Duc De La Rochefoucauld (1613–1680)
“I cannot give you the formula for success, but I can give you the formula for failure—which is: Try to please everybody.”
—Herbert B. Swope (1882–1958)