As painful as it is to think about Sirius spending 13 birthdays in azkaban, I would like to present the pain of what should of been 13 birthday's for Regulus.
Sirius had to deal with the flashbacks the dementors forced on him for years. In the beginning, there was no way of escaping the memory of his mother's yelling and father's beatings. No way to escape the memory of walking into his house of comfort, now ruins, to find his (platonic) soul mate's body on the floor, eyes wide open.
It wasn't until one particular flashback, that he was able to pull himself out. Sirius had ran away from home, and was staying at James'. To everyone's suprise, Regulus came to the door. Ignoring, are possibly not noticing Regulus' bags in his arms and frightened face, Sirius slammed the door and told him that he couldn't get Sirius to go back home.
It wasn't until Sirius pulled himself out of the flashback that he realized who those bags were for. They certainly weren't his, as the Black's probably burned his stuff as soon as the opportunity presented itself. In a state of shock, Sirius realized that those bags were... Regulus'. He came to Sirius in hopes that he could join him too.
Guilt over took Sirius, so much so that he hoped the flashbacks would start coming back. When they did, they had changed. Instead of violent memories of his parents, he saw flashbacks of his brother. Playing together at a young age, Sirius defending him from their father, and the more painful memories of their time in Hogwarts.
Through the flashbacks, Sirius started to realize how much Regulus paid attention to him when they were at Hogwarts, and the glare he always gave James. Sirius realized that James was everything Regulus used be, before he was sorted into Gryffindor.
The years of denial that Regulus didn't cared for him came crashing back, and eventually dissolved into overwhelming guilt. Regulus had tried to run away just like his brother, and it was because Sirius closed the door in his face that Regulus went on to becoming a Death Eater, forced to follow his parents footsteps. It was because of Sirius that Regulus willing partook in the Dark Lord's bidding, and died for it.
It wasn't until Sirius passed through the viel that he was able to apologize to Regulus. After being bombarded with hugs and a lengthly conversation with his favourite Potters, he noticed a small figure standing behind them. Regulus.
Before Reg could say anything, Sirius hugged him. The Blacks weren't known for physical (or any form) affection. This was Reg's first hug since before the sorting. There's a moment of silence between the two, then Regulus hugs him back.
Both are crying at this point when Sirius tells Reg of the years he spent thinking of him. He told him that he's sorry he didn't let him through the door, that he's sorry he couldn't be the big brother he needed, and he's sorry he drove Regulus to the dark side.
More silence follows before Regulus told Sirius how he actually died. He tells him of his adventure with Kreacher, and the promise they made not to tell anyone how he died. "Guess that's what I get for trying to be like you, Siri. Though I can see that didn't turn out so well for the both of us." Reg would tease.
For the first, Sirius was proud to be a Black.