## Yarra Gnisrever

### Problem 680

Let $N$ and $K$ be two positive integers.

$F_n$ is the $n$-th Fibonacci number: $F_1 = F_2 = 1$, $F_n = F_{n - 1} + F_{n - 2}$ for all $n \geq 3$.

Let $s_n = F_{2n - 1} \mod N$ and let $t_n = F_{2n} \mod N$.

Start with an array of integers $A = (A[0], \cdots, A[N - 1])$ where initially every $A\text{[}i]$ is equal to $i$. Now perform $K$ successive operations on $A$, where the $j$-th operation consists of reversing the order of those elements in $A$ with indices between $s_j$ and $t_j$ (both ends inclusive).

Define $R(N,K)$ to be $\sum_{i = 0}^{N - 1}i \times A\text {[}i]$ after $K$ operations.

For example, $R(5, 4) = 27$, as can be seen from the following procedure:

Initial position: $(0, 1, 2, 3, 4)$

Step 1 - Reverse $A[1]$ to $A[1]$: $(0, 1, 2, 3, 4)$

Step 2 - Reverse $A[2]$ to $A[3]$: $(0, 1, 3, 2, 4)$

Step 3 - Reverse $A[0]$ to $A[3]$: $(2, 3, 1, 0, 4)$

Step 4 - Reverse $A[3]$ to $A[1]$: $(2, 0, 1, 3, 4)$

$R(5, 4) = 0 \times 2 + 1 \times 0 + 2 \times 1 + 3 \times 3 + 4 \times 4 = 27$

Also, $R(10^2, 10^2) = 246597$ and $R(10^4, 10^4) = 249275481640$.

Find $R(10^{18}, 10^6)$ giving your answer modulo $10^9$.