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Concatenation Coincidence

 Published on Saturday, 13th March 2021, 10:00 pm; Solved by 479

Problem 751

A non-decreasing sequence of integers $a_n$ can be generated from any positive real value $\theta$ by the following procedure: \begin{align} \begin{split} b_1 &= \theta \\ b_n &= \left\lfloor b_{n-1} \right\rfloor \left(b_{n-1} - \left\lfloor b_{n-1} \right\rfloor + 1\right)~~~\forall ~ n \geq 2 \\ a_n &= \left\lfloor b_{n} \right\rfloor \end{split} \end{align} Where $\left\lfloor . \right\rfloor$ is the floor function.

For example, $\theta=2.956938891377988...$ generates the Fibonacci sequence: $2, 3, 5, 8, 13, 21, 34, 55, 89, ...$

The concatenation of a sequence of positive integers $a_n$ is a real value denoted $\tau$ constructed by concatenating the elements of the sequence after the decimal point, starting at $a_1$: $a_1.a_2a_3a_4...$

For example, the Fibonacci sequence constructed from $\theta=2.956938891377988...$ yields the concatenation $\tau=2.3581321345589...$ Clearly, $\tau \neq \theta$ for this value of $\theta$.

Find the only value of $\theta$ for which the generated sequence starts at $a_1=2$ and the concatenation of the generated sequence equals the original value: $\tau = \theta$. Give your answer rounded to 24 places after the decimal point.