By Lemma 3.13, there is an integer sequence,, such that as and uniformly on as.
Specifically, suppose that ∥ x ∥ = ⌊ x + 1 2 ⌋ (the nearest integer function), and { v n } n ≥ 0 is an integer sequence satisfying the recurrence formula v n = a 1 v n − 1 + a 2 v n − 2 + ⋯ + a s v n − s ( s ≥ 2 ).
That is, they studied the computational problem of the nearest integer function of ( ∑ k = n ∞ 1 u k ) − 1 and proved an interesting conclusion: ∥ ( ∑ k = n ∞ 1 u k ) − 1 ∥ = u n − u n − 1 for all n > n 0, where ∥ ⋅ ∥ denotes the nearest integer, namely ∥ x ∥ = ⌊ x + 1 2 ⌋, { u n } n ≥ 0 is an integer sequence satisfying the recurrence formula u n = a u n − 1 + u n − 2 + ⋯ + u n − s ( s ≥ 2 ).
In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one.
Let be an integer, a sequence of real numbers.
Let,, be an integer, and a sequence of real numbers such.
Let be any integer sequence such that and as.
We study the Manhattan Sequence Consensus problem (MSC problem) in which we are given k integer sequences, each of length ℓ, and we are to find an integer sequence x of length ℓ (called a consensus sequence) such that the maximum Manhattan distance of x from each of the input sequences is minimized.
This sequence provides an example showing that double-exponential growth is not enough to cause an integer sequence to be an irrationality sequence.
Let be a positive integer sequence such that,, and uniformly on as, where is any compact subset in and.
