Guess the next term in the following sequence:

π/2, π/2, π/2, π/2, π/2, π/2, π/2, ?

What's that you say? π/2? Ahahaha. No, I'm sorry, nice try, but the answer is

π(1/2 - 6879714958723010531/93561584944064090731

According to the mathoverflow comment, this seems to have been originally reported as a bug in mathematica, but it was later discovered that the math involved (an integral such that if you plug in N=0,1,2,3,4,5,6 you get π/2, but for N=7 you get the above monster) is

*actually that weird*.

This is like a way more cruel puzzle than the other high-watermark weird-yet-ultimately-sorta-reasonable sequence puzzle that I know of, which is:

Guess the next term in the following sequence:

0, 1, 2, 720!, ?

(where the ! is indeed a factorial)