Conjectured diagonal Ramsey numbers

Question

While doing some reading on the Wikipedia page for Ramsey numbers, I stumbled upon OEIS sequence A120414, which lists the allegedly conjectured values of the diagonal Ramsey numbers $R(n,n)$.

I haven't found any indications in the standard web resources (WP, Mathworld) that suggest that any of the numbers in the known ranges for $R(n,n)$, $n\geq 5$ is more probable than others.

What is this conjectured list based on? Considering the age of references on the OEIS page, the answer is probably elsewhere.

Answer 1

As your link informs, the numbers are given by:

$$a(n) = \lceil\left(3/2\right)^{(n-3)} \cdot n(n-1)\rceil$$

Which is in accordance with the known numbers $R(1,1),\dots,R(5,5)$ and fall within the limits given here (although there is an even better upper estimate found recently). However, the numbers on your link are not explicitly said to be more likely than others and might be conjectured here solely based on the formula above. I know of no reason why one "guess" of $R(n,n)$ should be bether than any other.

Answer 2

I imagine the conjecture is due to the author, who I believe to be the late Jeffrey Robert Boscole. Note that the formula is $$ a(n) = \left\lceil (3/2)^{n-3}\cdot n(n-1) \right\rceil $$ rather than $$ a(n) \stackrel{?}{=} \left\lceil (3/2)^{(n-3)n(n-1)} \right\rceil. $$