Is Lightroom still installed? If so, you can find your serial number at the location given in this document:
Suggestion: from now on, you need to store ALL of your serial number (not just Lightroom) in a location that you know you can find at a later time. What I do is I create text file on my computer that is named serial_numbers.txt, and place all of your serial numbers into this file; and I make backups to another disk and to "the cloud", so even if my house burns down, I still have access to all of my serial numbers.
I normally do keep the box but it has been misplaced in the chaos of life, I like the idea of putting it in notes, very smart, thanks for the tip!
Now I tried what the website says but when I get to the "Lightroom" stage of the instructions (see below)
/Users/[user name]/Library/Application Support/Adobe/Lightroom/Lightroom 6.0 Registration
There is no folder called Lightroom 6.0 Registration
Is your computer Windows or Mac?
Macbook pro from 2013
As I am not a Mac person, the only advice I can give you is to read the instructions carefully.
Particularly the part where it says:
Note: The user Library file is hidden by default on Mac OS X 10.7 and later. For temporary access to the user Library file, press Option and choose Go > Library in the Finder.