cp's OEIS Frontend

The sequence is nontrivial if and only if a(1) > 0. a(n) <= n for n <= 10000, but it is not known if this holds for all n. a(n) + a(n+1) <= n is usually but not always true (first exception is at n=509; a(509) + a(510) = 248 + 311 = 559).

For n > 1, a(n) = 0 if and only if a(n-1) is a record novel term, whereas every non-record novel term is followed by a nonzero term. Let S(n) be the set of unused terms prior to a(n), then step function |S(n)| increments +1 at a(k+1), where a(k) is a novel term. S(n) typically contains multiple copies of each unused number, providing a continuously incremented supply of least prior terms to add to repeat leading terms as the sequence extends. This suggests that there is always a next record, and hence that zero occurs infinitely many times. Indices of records: 1, 5, 10, 16, 24, 29, 35, 49, 54, 60, 66, 80, 86, 114, 136, 166, 176, 192, 198, 231, ...

If a(k) is a record term, we see a(k), 0, m, m, ... where m is the least member of S(k). Between any consecutive pair of zeros we see either no novel terms, in which case the trajectory climbs quickly to the next record term, or there are novel terms, each of which disturbs and extends the trajectory to the next record (see plots).