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:
“We are closed in, and the key is turned
On our uncertainty;”
—William Butler Yeats (18651939)
“Our ideas are the offspring of our senses; we are not more able to create the form of a being we have not seen, without retrospect to one we know, than we are able to create a new sense. He whose fancy has conceived an idea of the most beautiful form must have composed it from actual existence.”
—Henry Fuseli (17411825)
“My formula for greatness in human beings is amor fati: that one wants to change nothing, neither forwards, nor backwards, nor in all eternity. Not merely to endure necessity, still less to hide itall idealism is mendacity in the face of necessitybut rather to love it.”
—Friedrich Nietzsche (18441900)