[seqfan] Re: nice new board game puzzle A337663

[Neil Sloane]

I just finished updating A337663 with all the new results.  We still don't
know a(5)!

On Thu, Oct 8, 2020 at 7:04 AM <hv at crypt.org> wrote:

> Peter Kagey <peterkagey at gmail.com> wrote:
> :I posted this question to Code Golf Stack Exchange, and Arnauld Chevallier
> :confirmed the given terms:
> https://codegolf.stackexchange.com/a/212164/53884
>
> I don't see Arnauld's response there, do I need to do something special
> to see it?
>
> :My suspicion is that it will be hard to get more many terms because of
> :how quickly the search space grows.
> :
> :This is a really lovely sequence. I'd be interested to know more about
> this,
> :such as:
> :What is the 'widest' such a game can get? (The above example uses 13
> columns.)
>
> This gets quite tricky: for n >= 5 we can have multiple independent
> starting
> clusters, and it is not obvious to me how far away from each other those
> can be.
>
> Starting clusters can be independent only if they are the seed for the
> placement of one of the numbers 2 to 8, so there can be no more than 7
> of them. But that independence will make it a whole lot more difficult
> to calculate for n >= 5.
>
> My inclination is to keep them independent, and at each step consider
> each possible way of placing two or more of them in relation to each
> other (consistent with prior information) that might generate another
> possible location for the next integer to be placed. (Taking care,
> therefore, that no existing entry > 1 acquires a new neighbour. But
> two 1s _can_ become neighbours.)
>
> That sounds quite tough to code, and even tougher to make fast.
>
> Hugo
>
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