Probability of selecting two blue discs back to back

[BMAD]

If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, P(BB) = (15/21)×(14/20) = 1/2.

The next such arrangement, for which there is exactly 50% chance of taking two blue discs at random, is a box containing eighty-five blue discs and thirty-five red discs.

By finding the first arrangement to contain over 10¹² = 1,000,000,000,000 discs in total, determine the number of blue discs that the box would contain.

[Quantum.Mechanic]

OK, I thought that would be a good little puzzle program to write. I'm sure there's an easier way to figure this on paper though, waiting for y'all to clue me in. The first few elements led me to the following:

Spoiler

http://oeis.org/A046090, "a(a-1) = 2b(b-1)". The first element larger than 1e12 is 1070379110497.

 

[plasmid]

Spoiler

The picture in the spoiler above is an explanation with formulas that wouldn't look very nice if I tried to just write them out in text.

Edited September 2, 2017 by plasmid
changed to solve for the number of blue discs instead of the total number of discs