CF1692G.2^Sort

Given an array $a$ of length $n$ and an integer $k$ , find the number of indices $1 \leq i \leq n - k$ such that the subarray $[a_i, \dots, a_{i+k}]$ with length $k+1$ (not with length $k$ ) has the following property:

• If you multiply the first element by $2^0$ , the second element by $2^1$ , ..., and the ( $k+1$ )-st element by $2^k$ , then this subarray is sorted in strictly increasing order.

More formally, count the number of indices $1 \leq i \leq n - k$ such that $$$$2^0 \cdot a_i < 2^1 \cdot a_{i+1} < 2^2 \cdot a_{i+2} < \dots < 2^k \cdot a_{i+k}. $$$$

The first line contains an integer $t$ ( $1 \leq t \leq 1000$ ) — the number of test cases.

The first line of each test case contains two integers $n$ , $k$ ( $3 \leq n \leq 2 \cdot 10^5$ , $1 \leq k < n$ ) — the length of the array and the number of inequalities.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ( $1 \leq a_i \leq 10^9$ ) — the elements of the array.

The sum of $n$ across all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output a single integer — the number of indices satisfying the condition in the statement.

In the first test case, both subarrays satisfy the condition:

• $i=1$ : the subarray $[a_1,a_2,a_3] = [20,22,19]$ , and $1 \cdot 20 < 2 \cdot 22 < 4 \cdot 19$ .
• $i=2$ : the subarray $[a_2,a_3,a_4] = [22,19,84]$ , and $1 \cdot 22 < 2 \cdot 19 < 4 \cdot 84$ .

In the second test case, three subarrays satisfy the condition: - $i=1$ : the subarray $[a_1,a_2] = [9,5]$ , and $1 \cdot 9 < 2 \cdot 5$ .

• $i=2$ : the subarray $[a_2,a_3] = [5,3]$ , and $1 \cdot 5 < 2 \cdot 3$ .
• $i=3$ : the subarray $[a_3,a_4] = [3,2]$ , and $1 \cdot 3 < 2 \cdot 2$ .
• $i=4$ : the subarray $[a_4,a_5] = [2,1]$ , but $1 \cdot 2 = 2 \cdot 1$ , so this subarray doesn't satisfy the condition.