A Recursively Defined Sequence

Given is the function $f(x) = \lfloor 2^{30.403243784 – x^2}\rfloor \times 10^{-9}$ ($\lfloor \, \rfloor$ is the floor-function),
the sequence $u_n$ is defined by $u_0 = -1$ and $u_{n + 1} = f(u_n)$.
Find $u_n + u_{n + 1}$ for $n = 10^{12}$.
Give your answer with $9$ digits after the decimal point.

First, let’s understand what we’re working with.

We have a function f(x) which is defined as the product of $\lfloor 2^{30.403243784 – x^2}\rfloor$ (that is, the greatest integer less than or equal to $2^{30.403243784 – x^2}$), and $10^{-9}$.

Then, we have a sequence $u_n$ which is initially set at $u_0 = -1$ and subsequently defined as $u_{n+1} = f(u_n)$.

We are asked to find $u_n + u_{n + 1}$ for $n = 10^{12}$ with nine decimal digits.

Now, while it’s true that this is a sequence defined over a very large number of terms, it’s also a function that defines the sequence that may recycle or repeat a period. Identifying such a pattern could greatly simplify the problem.

So, let’s first check the initial values in the sequence:

– $u_0 = -1$

– $u_1 = f(u_0) = f(-1) = \lfloor 2^{30.403243784 – (-1)^2}\rfloor \times 10^{-9} = \lfloor 2^{30.403243784 – 1}\rfloor \times 10^{-9} = 0.7052308467$

– $u_2 = f(u_1) = f(0.7052308467) = \lfloor 2^{30.403243784 – (0.7052308467)^2}\rfloor \times 10^{-9} = 1.571008042$

– $u_3 = f(u_2) = f(1.571008042) = \lfloor 2^{30.403243784 – (1.571008042)^2}\rfloor \times 10^{-9} = 0.7052308467$

Notice that here we have found a pattern. $u_1$ and $u_3$ are equal. This means that the following terms will repeat the pattern.

This is a loop with length $2$ and is as follows: $0.7052308467$ and $1.571008042$. So instead of finding series up to $n = 10^{12}$, we can simply observe that the sum will repeat every two terms. Since $10^{12}$ is even, we will have an equal number of sequences of $0.7052308467$ and $1.571008042$. Hence the sum of $u_n$ and $u_{n+1}$ will simply be the sum of these two.

So $u_{10^{12}} + u_{10^{12} + 1} = 0.7052308467 + 1.571008042 = 2.276238889$.

The nine-decimal-digit answer is 2.276238889.

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