Question

Find the number of subsets of the set of n consecutive positive integers, {1,2,...,n} that do not contain two elements whose difference is k, where k≤n−1
Someone sent me a proof of it similar to some ideas that were discussed before. I will post the proof in the post below.

Answer 1

consider a set {1, 2, 3, 4, 5}

for k=2, you need to find number of subsets of this set such that no two elements in the subset have a difference of 2

example subset :
{1,2,5}

Answer 2

okay

Answer 3

so k right now is everything below n-1 right in his question

Answer 4

and equal to n-1