Why is it always a conspiracy? Flip a quarter 100 million times, you'll eventually get heads 100 times in a row. Now flaw in the quarter, just as it's not a flaw that a few blocks get solved in a short period.
You need some serious probability classes around here. No offense please...it is a joke...
If you flip a quarter 100 million times and the probability of getting sometime 100 times heads in a row is 10^8 / 2^100 x(10^8-100) =7.8 10^{-15} pretty small
In order to get heads 100 times in a row you need to flip your quarter about 10^22 times...my guess is that your quarter will disintegrate in the process...
It's slightly more complicated as it involves Bayesian probability although you're close. A run of 20 heads is a roughly 1 in a million occurrence and the required rolls goes up exponentially the higher you get. For more information, see:
http://www.drdobbs.com/architecture-and-design/20-heads-in-a-row-what-are-the-odds/229300217
Your guy in the blog is enumerating all possibilities of coin flips. In order to have the actual probability for the problem it is much simpler: You start flipping your coin and you stop when you end up with a streak of n=100 or reach the maximum number of throws N=10^8. The starting point of your n=100 streak can be any point between 1 and N-100=10^8-100. The probability streak of n consecutive heads is 1/2^n and the probability to reach the k-throw is about 1 (it is 1-(prob of finding the streak before we reach k)) and this explains my formula, which is accurate except for the slight approximation. You only need the recursive analysis if n is much smaller (or N much larger).