Let's suppose that we know where the first occurrence of a sequence of n digits appears in the number pi decimal section. For example, the first occurrence of the sequence 26091955 appears in the decimal position 107,607,919.
Is there any way to calculate manually (pen and paper) the decimal digits that would follow? (in the example, it'd be 25916325541684811916)
Apparently there is a way to calculate the N-th digit of $\pi$ (in base 16) independent of the other digits.
So, technically the answer is YES. but it's even better than your question. I don't care about the sequence of digits, you just tell me the position.
The formula is (taken from the link mentioned above)
$$\pi=\sum_{k=0}^{\infty}16^{-k} \times\left(\frac{4}{8k+1} - \frac{2}{8k +4} - \frac{1}{8 k +5} - \frac{1}{8k +6}\right).$$
Source: Su, Francis E., et al. "Finding the N-th digit of Pi." Math Fun Facts. http://www.math.hmc.edu/funfacts.