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.
For me it was a favorite place to visit every so often. I also really enjoyed mathworld.wolfram.com a few decades ago. (A true shame that he went insane)
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.
That sequence is not known to match what you asked for:
>> Conjecturally, every positive integer occurs in the sequence or one of its n-th differences, which would imply that the sequence and its n-th differences partition the positive integers.
For an intuition of why this might be hard to prove, note that you had to insert 7 into your structure before you inserted 5. In the general case, there might be a long waiting period before you're able to place some particular integer n. It might be infinitely long.
The initial sequence is 1, 3, 7, 12, 18, 26, 35, etc. The difference between each term in that sequence produces a second sequence: 2, 4, 5, 6, 8, 9, 10, etc. If you merge those two sequences together in sorted order, you get 1, 2, 3, 4, 5, 6, 7, etc. Each whole number appears in the result exactly once.
I wonder why the title of the sequence isn't set to "Hofstadter's sequence" since that seems to be what it's called according to A030124 when it refers back to this one
Hofstadter introduces several sequences in GEB, [0] may be an interesting submission on its own but I was especially captivated by this self-referencing one. Plus a title including both Hofstadter's sequence and a description is too long for HN and I preferred the descriptive one
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[ 2.4 ms ] story [ 32.5 ms ] threadhttps://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.
Something like
Oh, here it is: https://oeis.org/A035313That sequence is not known to match what you asked for:
>> Conjecturally, every positive integer occurs in the sequence or one of its n-th differences, which would imply that the sequence and its n-th differences partition the positive integers.
For an intuition of why this might be hard to prove, note that you had to insert 7 into your structure before you inserted 5. In the general case, there might be a long waiting period before you're able to place some particular integer n. It might be infinitely long.
[0] https://en.wikipedia.org/wiki/Hofstadter_sequence