[seqfan] Re: Nice nontrivial integer sequence

[Christian Sievers]

On Sat, Mar 11, 2023 at 07:58:01PM -0600, Tim Peters wrote:

> The page for A360477
> 
>     https://oeis.org/A360447
> 
> contains
> 
>     FORMULA a(n) = a(n-1) + a(n+1) if a(n+1) < a(n)
> 
> I expect "a formula" to show a way to compute a(n), but that's not what
> this is. It appears to be a claim about 3 consecutive terms in some cases,
> but it's not a true claim. I think the first place it fails is at n=18:

Well, I have been excited and then disappointed by such OEIS formula
entries before, so I feel you, but in general such entries are accepted.

> Alas, I can't think of a true claim related to what it says, so I can't
> suggest "a fix". I don't believe anyone has "a formula" for a(n) in the
> usual meaning of the word. Perhaps this section should just be deleted?

While I agree with its deletion, I want to note that there is a possible
fix: just strengthen the condition by adding "and a(n-1) < a(n)".

The proof needs to discuss the list in progress, so it's not reasonable
to talk about a(n) as the position of a number changes, instead let's 
claim that a number is the sum of its neighbours if both are smaller.
That's true when a number gets inserted between two numbers, and when it
is appended to the end of the list, then any number that appears right
to it will be bigger. Also, any number that appears as a new neighbour
to a number will be bigger than it, so the precondition is not
satisfied. (Much words about a triviality.)

But I wonder about a different question:
Do we know that each position eventually gets stable?
In other words: are we sure we have a well defined infinite series?


All the best
Christian