I allowed myself to edit the title slightly, to hopefully make it more clear to people familiar with OEIS - it was a celebratory contest to determine which sequence should be the 400'000th entry in the database.
This might be skirting the topicality guidelines a bit, but I think we've had enough OEIS-adjacent things posted here to justify this. Even if you don't find the contest sequences particularly interesting, hopefully you'll still think that this milestone is worth celebrating? :)
Understanding the winning sequence took me a moment - Sloane's drawing and the listed example for n=8 are helpful. I still don't understand the justification of why this sequence contains every natural number; hopefully I'll get it when I look at this at a more reasonable hour.
I think the argument goes something like this: after a tower that's taller than all the ones that came before it, the sequence "starts over" for a while, but with 1 added to account for the tall tower. The length of the repeat is proportional to sqrt(record-setting height), and towers get arbitrarily tall, so you repeat arbitrarily long initial segments of the sequence with 1 added. This means that if a(n) = k, then there's some "record-setting" m such that a(m+n) = k + 1.
I'm not sure if that works. Let's say we get to the tower of height 10 - at this point we still haven't seen [4..9], but this reasoning would only justify the existence of a tower of height 11. The subsequence after that tower won't differ much from the subsequence after towers of height 10:
[OP] dzwdz | 16 hours ago
I allowed myself to edit the title slightly, to hopefully make it more clear to people familiar with OEIS - it was a celebratory contest to determine which sequence should be the 400'000th entry in the database.
This might be skirting the topicality guidelines a bit, but I think we've had enough OEIS-adjacent things posted here to justify this. Even if you don't find the contest sequences particularly interesting, hopefully you'll still think that this milestone is worth celebrating? :)
Understanding the winning sequence took me a moment - Sloane's drawing and the listed example for n=8 are helpful. I still don't understand the justification of why this sequence contains every natural number; hopefully I'll get it when I look at this at a more reasonable hour.
quasi_qua_quasi | 2 hours ago
I think the argument goes something like this: after a tower that's taller than all the ones that came before it, the sequence "starts over" for a while, but with 1 added to account for the tall tower. The length of the repeat is proportional to sqrt(record-setting height), and towers get arbitrarily tall, so you repeat arbitrarily long initial segments of the sequence with 1 added. This means that if a(n) = k, then there's some "record-setting" m such that a(m+n) = k + 1.
[OP] dzwdz | 24 minutes ago
I'm not sure if that works. Let's say we get to the tower of height 10 - at this point we still haven't seen [4..9], but this reasoning would only justify the existence of a tower of height 11. The subsequence after that tower won't differ much from the subsequence after towers of height 10:
is an example from early on in the sequence.