Project Euler Problem 751
A non-decreasing sequence of integers an can be generated from any positive real value theta by the following procedure:
Solution
Answer: 2.223561019313554106173177
Let $T(\theta)$ denote the concatenation map generated by the sequence:
$$a_1=\lfloor \theta \rfloor,\qquad b_1=\theta,\qquad b_n=\lfloor b_{n-1}\rfloor\bigl(b_{n-1}-\lfloor b_{n-1}\rfloor+1\bigr), \qquad a_n=\lfloor b_n\rfloor.$$
We seek the unique fixed point satisfying:
- $a_1 = 2$,
- $\tau = 2.a_2a_3a_4\ldots = \theta$.
A robust way to solve this is fixed-point iteration:
- Start with a decimal beginning with
2.. - Generate the sequence $a_n$ from the current approximation.
- Form the new decimal by concatenating those $a_n$.
- Repeat until the digits stabilize.
The iteration converges to the unique self-consistent value:
$$\theta = 2.22356101931355410617317719528048680814072321360554\ldots$$
Rounding to 24 places after the decimal point gives:
Answer: 2.223561019313554106173177