Question

I have to make a non-continuous integer sequence a_1, a_2, ..., a_n which satisfies

{a_1, a_2, ..., a_n} ∈ [1, n]

|a_i - a_(i-1)| > 1

How to make it? I appreciate you a hint or some help. Thanks 🙂

Answer 1

When there’re less than 4 elements, it is impossible to do.

When there’re 4 or more elements, we can construct the sequence like this:

if n is even:

[n-1, n-3, ... , 1, n, n-2, ..., 2]

if n is odd:

[n, n-2, ..., 1, n-1, n-3, ..., 2]

Now that n is equal or large than 4, |1-n| = |n-1| >= 3, |1-(n-1)| = |1-n+1| = |n| >= 4, the sequence satisfies the constraint.