Thought Process

A complete brute force solutions, surprisingly most solutions in the thread were also brute force.

Here's an outline of what I did:

1. Start by finding all n's for a^2 + 7b^2

   1. Note that a < sqrt(2,000,000,000), then restrict b accordingly to a

   2. Looped through all possible a, b and put all a^2 + 7b^2 in a set called sqs7

2. Find all n's for a^2 + 3b^2

   1. Note that a < sqrt(max(sqs7)), then restrict b accordingly to a

   2. Looped through all possible a, b and put all a^2 + 3b^2 that also appeared in sqs7 in a set called sqs3 

3. Find all n's for a^2 + 2b^2

   1. Note that a < sqrt(max(sqs3)), then restrict b accordingly to a

   2. Looped through all possible a, b and put all a^2 + 2b^2 that also appeared in sqs3 in a set called sqs2 

4. Find all n's for a^2 + b^2

   1. Note that a < sqrt(max(sqs2)), then restrict b accordingly to a

   2. Looped through all possible a, b and put all a^2 + b^2 that also appeared in sqs2 in a set called sqs1 (squares 7)

5. Then the length of sqs1 is the answer.

This code worked for 1,000,000,000 but to compute the final answer I had memory issues, so I just implemented a segmented version of the above and let it run for ~18 mins and voila.

For the segmented version you just have to keep track for each a what was the maximum b. Then in the next segment, you just start from that b. 

I selected the segment size to be 2,000,000 but running it a few times and guessing a good segment size.