I'm really sorry to hear about Max...and trust me, I know where you're coming from. I went through the exact same thing last year when one of my dogs, a German shorthaired pointer named Marta, had to be put to sleep.
She was 13 years, seven months old, so she'd had a good run, but it was really, really rough when she went. I mean, we'd gotten her when I was eight years old, so she and I grew up together - and in the PDA section of my high school yearbook, there's a photo of me and her.
And it was the same with us - the vet sent us a sympathy card a few days later.
There's a group out there called Rainbow Bridge, and they've got a series of lists of the names of pets who've gone on to the next life. Maybe you could put Max's name on one of the lists?