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:
“Because you live, O Christ,
the spirit bird of hope is freed for flying,
our cages of despair no longer keep us closed and life-denying.
The stone has rolled away and death cannot imprison!
O sing this Easter Day, for Jesus Christ has risen!”
—Shirley Erena Murray (20th century)
“While they stand at home at the door he is dead already,
The only son is dead.
But the mother needs to be better,
She with thin form presently drest in black,
By day her meals untouch’d, then at night fitfully sleeping, often waking,
In the midnight waking, weeping, longing with one deep longing,
O that she might withdraw unnoticed, silent from life escape and
withdraw,
To follow, to seek, to be with her dear dead son.”
—Walt Whitman (1819–1892)
“I feel like a white granular mass of amorphous crystals—my formula appears to be isomeric with Spasmotoxin. My aurochloride precipitates into beautiful prismatic needles. My Platinochloride develops octohedron crystals,—with a fine blue florescence. My physiological action is not indifferent. One millionth of a grain injected under the skin of a frog produced instantaneous death accompanied by an orange blossom odor.”
—Lafcadio Hearn (1850–1904)