Ph.D. Physics graduate from the University of Illinois, June, 1969.
I applying to register only to comment on on sequence A006454, i.e triangular numbers which are one less than a square. My comment will point out that a(n) is the number of object balls (triangular because, in the game, the balls are "racked" in a triangular frame) in a pocket billiards set that can, when the cue ball is added, be compactly shipped in a square box. Thus in my pocket billiard game the 15 object balls were neatly shipped, with the cue ball, in a 4 ball by 4 ball square box.
I think it is nice to have a physical example of why having a triangular number being one less than a square number is useful.