cube triangular numbers and intersections of sequences

[Tanya Khovanova]

Hello Seqfans,

Recently I've been interested in intersections of famous sequences. (like all even perfect numbers are triangular).

I can't find a cube triangular number other than 1. Is there a proof that they do not exist?
I can easily prove that cube pronic numbers do not exist (pronic numbers are twice triangular numbers).

I also tried to submit some intersections of sequences, but there should be a limit to this. 

What is the basic list of sequences such that all intersections of those sequences deserve to be in the database?

Best, Tanya