A14901. Arithmetic Operations

You are given an array of integers $a_1, a_2, \ldots, a_n$ .

You can do the following operation any number of times (possibly zero):

- Choose any index $i$ and set $a_i$ to any integer (positive, negative or $0$ ).

What is the minimum number of operations needed to turn $a$ into an arithmetic progression? The array $a$ is an arithmetic progression if $a_{i+1}-a_i=a_i-a_{i-1}$ for any $2 \leq i \leq n-1$ .

The first line contains a single integer $n$ ( $1 \le n \le 10^5$ ).

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 10^5$ ).

Print a single integer: the minimum number of operations needed to turn $a$ into an arithmetic progression.

In the first test, you can get the array $a = [11, 10, 9, 8, 7, 6, 5, 4, 3]$ by performing $6$ operations:

- Set $a_3$ to $9$ : the array becomes $[3, 2, 9, 8, 6, 9, 5, 4, 1]$ ;
- Set $a_2$ to $10$ : the array becomes $[3, 10, 9, 8, 6, 9, 5, 4, 1]$ ;
- Set $a_6$ to $6$ : the array becomes $[3, 10, 9, 8, 6, 6, 5, 4, 1]$ ;
- Set $a_9$ to $3$ : the array becomes $[3, 10, 9, 8, 6, 6, 5, 4, 3]$ ;
- Set $a_5$ to $7$ : the array becomes $[3, 10, 9, 8, 7, 6, 5, 4, 3]$ ;
- Set $a_1$ to $11$ : the array becomes $[11, 10, 9, 8, 7, 6, 5, 4, 3]$ .

$a$ is an arithmetic progression: in fact, $a_{i+1}-a_i=a_i-a_{i-1}=-1$ for any $2 \leq i \leq n-1$ .

There is no sequence of less than $6$ operations that makes $a$ an arithmetic progression.

In the second test, you can get the array $a = [-1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38]$ by performing $10$ operations.

In the third test, you can get the array $a = [100000, 80000, 60000, 40000, 20000, 0, -20000, -40000, -60000, -80000]$ by performing $7$ operations.