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:
“Night hath closed all in her cloak,
Twinkling stars love-thoughts provoke,
Danger hence good care doth keep,
Jealousy itself doth sleep;”
—Sir Philip Sidney (1554–1586)
“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)
“Hidden away amongst Aschenbach’s writing was a passage directly asserting that nearly all the great things that exist owe their existence to a defiant despite: it is despite grief and anguish, despite poverty, loneliness, bodily weakness, vice and passion and a thousand inhibitions, that they have come into being at all. But this was more than an observation, it was an experience, it was positively the formula of his life and his fame, the key to his work.”
—Thomas Mann (18751955)