Before the current lower bound for tree(3) in the weak tree function, there was a lower bound of 262140. How did they reach that number?

Question

Before the current lower bound for tree(3) in the weak tree function (844,424,930,131,960), there was a lower bound of 262140. How did they reach that number? From my searches it seems like there's really nothing available online about that.

Answer 1

Finally I found the original post of this lower bound. I used a little different approach to calculate the number, which I believe is simpler.

Here is the list of trees:
1:(()()())
2:((())(()))
3:(((())())())
4:((((()())))())
5:(((((()()))())))
6:((((()()))()))
7:(((()()))())
8:((((((((()())())))))))
9:(((((((()())()))))))
10:((((((()())())))))
11:(((((()())()))))
12:((((()())())))
13:(((()())()))
14:((()())())
15:((((((((((((((((())))))))))))))))())

The 15th tree has two children, one is a chain of length 16 and the other is single. Now we delete the 16th of the chain and we add two to the root, like this:
16:(((((((((((((((((()))))))))))))))())))

Now we delete these added nodes one at a time, so tree 18 will be like tree 15 but with a chain of 15. At tree 19 we can add six (3+19-15-1) to the root, and we delete them one at a time till tree 25 a chain of 14 without any added nodes at the root. The general formula is 2ⁿ⁺²-n-4 where n is the length of the chain. So after 2¹⁶⁺²-16-4=262124 we have deleted the entire chain and we have (()). Now there's one more to go.

We started with tree 15. So the formula is 15+262124+1=262140.