751. Concatenation Coincidence

A non-decreasing sequence of integers $a_n$$a_n$ can be generated from any positive real value $\theta$$\theta$ by the following procedure:

Where $\lfloor \cdot \rfloor$$\left\lfloor \cdot \right\rfloor$ is the floor function.

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

The concatenation of a sequence of positive integers $a_n$$a_n$ is a real value denoted $\tau$$\tau$ constructed by concatenating the elements of the sequence after the decimal point, starting at $a_1$$a_1$: $a_1.a_2{a}_3{a}_4...$$a_1.a_2a_3a_4...$

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

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

751. 拼接相合

下述过程能使用任意正实数 $\theta$$\theta$ 生成一个不减的整数序列 $\{a_n\}$$\{a_n\}$：

其中 $\lfloor \cdot \rfloor$$\left\lfloor \cdot \right\rfloor$ 是取整函数。

例如，取 $\theta =2.956938891377988...$$\theta=2.956938891377988...$ 就能产生斐波那契数列 $2,3,5,8,13,21,34,55,89,...$$2, 3, 5, 8, 13, 21, 34, 55, 89, ...$。

对任意正整数序列 $\{a_n\}$$\{a_n\}$，定义其 拼接数 为：把 $a_1$$a_1$ 作为整数部分，在小数点后顺次写下序列的其他元素所得到的实数 $\tau =a_1.a_2{a}_3{a}_4...$$\tau = a_1.a_2a_3a_4...$。

例如，由 $\theta =2.956938891377988...$$\theta=2.956938891377988...$ 生成的斐波那契数列的拼接数为 $\tau =2.3581321345589...$$\tau=2.3581321345589...$，显然对这个 $\theta$$\theta$ 有 $\tau \ne \theta$$\tau \neq \theta$。

求 $\theta$$\theta$ 的唯一值，使其满足 $a_1=2$$a_1 = 2$ 且 $\theta$$\theta$ 生成的序列的拼接数 $\tau$$\tau$ 等于 $\theta$$\theta$。将答案四舍五入至小数点后第 $24$$24$ 位。

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