Why would someone ever do that?

Once, on a rainy Friday, I went back from school, got home and had lunch. After that, I got really bored while watching TV, so I decided to build towers with my 504 wooden blocks. After some minutes, I realized that I was able to build a huge number of towers using my wooden blocks. So, in order to reduce this number, I only built towers that had the same number of blocks. I liked so much building those towers that I decided to show them to my brother. However, I only built sequences of up to 3 towers to show him.

Let a be the sum of the number of towers of every possible sequence that could have been made by me, and let b be the number of blocks that I used to build each tower of all the possible sequences that was shown to my brother. What is the value of a + b?

Notes and Assumptions

  • A sequence of towers is a sequence of a certain number of towers with the same number of wooden blocks. Every sequence made by me had a total number of 504 blocks, in other words, I used all the wooden blocks in every sequence.

  • Consider that the space which I used to build the towers is infinite.


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