OEIS is such a wonderful reference. I've had occasions where software I was building needed to compute certain sequences, but I hadn't yet figured out the underlying math. I popped the sequence into OEIS and found the closed form solution. It was a huge productivity boost.

Like 'even and odd' on steroids.

Coding exercise: write a function

    boolean isInSequence(n):

that decides whether the given integer is part of that sequence or not. However, pre-storing the sequence and only performing a lookup is not allowed.

    rec ((x:xs),p) = (filter (/= p+x) xs,p+x)
    sequ = map snd $ iterate rec ([2..],1)

Recursive (n choose 2) is my favorite.

https://oeis.org/A086714

If you think about it, it quantifies emergence of harmonic interference in the superposition of 4 distinct waveforms. If those waveforms happen to have irrational wavelengths (wrt. each other), their combination will never be in the same state twice.

This obviously has implications for pseudorandomness, etc.

Is there a sequence where the sequence and all its differences contain each positive integer once?

Something like

    1 3 9   26  66
     2 6  17  40
      4 11  23
       7  12
        5

Oh, here it is: https://oeis.org/A035313