1787D · Game on Axis

Problem: There are $n$ points $1,2,\ldots,n$, each point $i$ has a number $a_i$ on it. You're playing a game on them. Initially, you are at point $1$. When you are at point $i$, take following steps:

• If $1\le i\le n$, go to $i+a_i$,
• Otherwise, the game ends.

Before the game begins, you can choose two integers $x$ and $y$ satisfying $1\le x\le n$, $-n \le y \le n$ and replace $a_x$ with $y$ (set $a_x := y$). Find the number of distinct pairs $(x,y)$ such that the game that you start after making the change ends in a finite number of steps.

Notice that you do not have to satisfy $a_x\not=y$.

Input Format: Each test contains multiple test cases. The first line contains an integer $t$ ($1\le t\le 10^4)$ — the number of test cases.

The first line of each test case contains one integer $n$ ($1\le n\le 2\cdot 10^5$) — the number of points.

The second line contains $n$ integers $a_1,a_2,\ldots,a_n$ ($-n \le a_i \le n$) — the numbers on the axis.

It's guaranteed that the sum of $n$ does not exceed $2\cdot 10^5$.

Output Format: For each test case, print a line containing a single integer — the number of distinct pairs $(x,y)$ with which the game ends.

Note: In the first test case, the pairs $(x,y)$ with which the game ends are $(1,-1)$ and $(1,1)$, corresponding to the routes $1\rightarrow 0$ and $1\rightarrow 2$. Note that $(1,2)$ is invalid since when $n=1$, $y=2$ violates $-n\le y\le n$. $(1,0)$ is also invalid since you will go from $1$ to $1$ forever.

In the second test case, the pairs are $(1,-2),(1,-1),(1,2),(2,-2),(2,-1),(2,0),(2,1),(2,2)$.

In the fourth test case, the pairs are $(1,-2),(1,-1),(1,1),(1,2),(2,-2),(2,1),(2,2)$.