One (probably) final tree update for now
Sep. 3rd, 2023 03:59 pmI updated the tree page to talk about what happens when you increase the additive constant!
By that I mean, what if, instead of n+L(n)=R(n), we did n+L(n)+c=R(n)? Actually, that's how I'll do things for the 0-rooted case; for the 1-rooted case I'll adjust things by one, say n+L(n)+(c-1)=R(n), so that it's just the 0-rooted version plus one.
Then c=1 is what we started with -- the 0-rooted version is the dual Zeckendorf tree, while the 1-rooted version is Beren's original tree, the Zeckendorf parity tree.
Meanwhile c=2 yields the other variants I discussed; the 0-rooted version is the dual Zeckendorf parity tree, while the 1-rooted version is the plain old Zeckendorf tree.
But what about c=3, and higher? Well, for c=3 I have a description, see the page. (Well, I have a fairly nice description for the 0-rooted version, not so much for the 1-rooted version; I'm going to stick to the 0-rooted versions for c≥3, I think.) But for c≥4... it's a mess. I tried c=4 and c=5 and don't see any nice description for either of them.
That said, despite this fact, these trees still appear to be full of Fibonacci patterns! Hopefully someone can figure out what's up with that, but I think I need to put this problem down for now -- I'm still so behind on other things...
By that I mean, what if, instead of n+L(n)=R(n), we did n+L(n)+c=R(n)? Actually, that's how I'll do things for the 0-rooted case; for the 1-rooted case I'll adjust things by one, say n+L(n)+(c-1)=R(n), so that it's just the 0-rooted version plus one.
Then c=1 is what we started with -- the 0-rooted version is the dual Zeckendorf tree, while the 1-rooted version is Beren's original tree, the Zeckendorf parity tree.
Meanwhile c=2 yields the other variants I discussed; the 0-rooted version is the dual Zeckendorf parity tree, while the 1-rooted version is the plain old Zeckendorf tree.
But what about c=3, and higher? Well, for c=3 I have a description, see the page. (Well, I have a fairly nice description for the 0-rooted version, not so much for the 1-rooted version; I'm going to stick to the 0-rooted versions for c≥3, I think.) But for c≥4... it's a mess. I tried c=4 and c=5 and don't see any nice description for either of them.
That said, despite this fact, these trees still appear to be full of Fibonacci patterns! Hopefully someone can figure out what's up with that, but I think I need to put this problem down for now -- I'm still so behind on other things...
