A15734. Game on Axis

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 $a_x\not=y$ .

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 $2\cdot 10^5$ .

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

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 $(1,-2),(1,-1),(1,1),(1,2),(2,-2),(2,1),(2,2)$ .