TITLE: How did Ramanujan discover this identity?
QUESTION [35 upvotes]: Let $$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and
$ad=bc$, then
$$64 F_6 F_{10}=45 F_8^2$$
This fascinating identity is due to Ramanujan and can be found in "Ramanujan for Lowbrows", by B.C. Berndt and S. Bhargava. Would anyone have any idea how Ramanujan discovered this identity?
The proofs of the identity offered so far in the papers "A Note on an Identity of Ramanujan", by T. S. Nanjundiah, "Two or Three Identities of Ramanujan", by M.D. Hirschhorn and "A remarkable identity found in Ramanujan's third notebook", by B.C. Berndt and S. Bhargava make the identity less mysterious, but how Ramanujan found the identity in the first place still remains a mystery. As Berndt and Bhargava remarked, it is also not clear whether this is an accidental, isolated result (along with the 3-7-5 counterpart discovered by Hirschhorn), or if there is  some deeper theorem lurking behind it.

REPLY [2 votes]: How Ramanujan found the identity is, of course, a mystery, but with the intelligent use of a computer algebra system, many such identities can be found. I wrote an article about this. Two of the simpler identities that were not listed above include
$$
F_{-2}F_3^2 = -3 F_{-1}^2 F_6
$$
and
$$
245F_3 F_{11} + 330F_7^2 = 539F_5 F_9 
$$