@athin has found the mathematical pattern in the sequence and created a useful general way to display it pictorially (go see their post and upvote). However, a slightly more accurate diagrammatic rendering would be as follows:
This is because the sequence reflects:
The significance of 23 March in the hints is that:
The pattern emerges as:
Why does the ...
Note: I have had this sat in the edit-box for about the last 18 hours, because I didn't feel that it was enough to post (since it conspicuously fails to explain the first two terms, and I'm pretty sure something more "concrete" is intended). But I see someone else has posted a partial with even less in it, so I'm posting this too. Of ...
Most of the work has already been done by @Florian F and @Gareth McCaughan. Based on their insights here is a geometric reading that I believe covers all cases:
Based on this I claim that the subsequent terms are
Re OP's latest hint:
If you like this answer and haven't already done so make sure to also check out Florian's and especially Gareth's answers ...
From Googling about a bit it seems like all the words are
In particular, the correspondence seems to be
Why did I run around crazy for after this? What's my problem?
No idea on the Bonus but maybe somebody else will get inspiration.
I think the length of the longest such sequence is
Proof of upper bound
Proof of lower bound
How did I find such a sequence
As Florian F has pointed out in the comments there is a simpler integer choice. Given that
And pushing this Fibonacci link a bit further we know that