[seqfan] A269783

[Frank Adams-Watters]

Can we find a proof of the comment in https://oeis.org/draft/A269783 that every ordered pair occurs? It seems obvious, and the numerical evidence is strong, but I have not been able to prove it.

It is trivially easy to show that the conjecture is true if every integer occurs infinitely often, and still quite easy to show that this follows if there are  infinitely many integers occurring infinitely often.

One approach to proving the latter would be to show that a(n) = o(n), perhaps O(sqrt(n)), but I don't see any way towards that result.

Alternatively, one could get the desired final result from careful analysis of the patterns in the sequence; but again, I don't see  how to get there.

Note that sequences with definitions like this can easily fail to include all integer pairs. For example, if the numbers following each n are 0, 1, -1, 2, -2, ..., independent of n we get the sequence 0, 0, 1, 0, -1, 0, 2, 0, -2, 0, .... If we always go up first (n, n+1, n-1, n+2, ...), we will get 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, .... Such problems usually manifest themselves right away - but that isn't a proof.

Franklin T. Adams-Watters

P.S. If people take this offline, please include me in the discussion.