If you read the demonstration carefully, you will see right at the beginning the ◊ 'operator' as I define it:

◊n=1+2+3+...+n

I just reduced your problem to a simpler version. Instead of trying to build an expression for the sum of first 2n naturals, I got the expression for the sum of first N naturals, ◊n. After that you just plug 2n: ◊(2n) and get the expression.

Think '◊n' as ◊(n) or f(n). There is nothing special about ◊.. It's just a symbol I usually associate with the 'function' that sums the 1st N naturals.

Now read my post 2 or 3 more times and try to catch up with a sheet of paper, line by line.