2145 - Count the Hidden Sequences.

Medium

You are given a 0-indexed array of n integers differences, which describes the differences between each pair of consecutive integers of a hidden sequence of length (n + 1). More formally, call the hidden sequence hidden, then we have that differences[i] = hidden[i + 1] - hidden[i].

You are further given two integers lower and upper that describe the inclusive range of values [lower, upper] that the hidden sequence can contain.

• For example, given differences = [1, -3, 4], lower = 1, upper = 6, the hidden sequence is a sequence of length 4 whose elements are in between 1 and 6 ( inclusive ).

Return the number of possible hidden sequences there are. If there are no possible sequences, return 0.

Example 1:

Input: differences = 1,-3,4, lower = 1, upper = 6

Output: 2

Explanation: The possible hidden sequences are:

• 3, 4, 1, 5

• 4, 5, 2, 6

Thus, we return 2.

Example 2:

Input: differences = 3,-4,5,1,-2, lower = -4, upper = 5

Output: 4

Explanation: The possible hidden sequences are:

• -3, 0, -4, 1, 2, 0

• -2, 1, -3, 2, 3, 1

• -1, 2, -2, 3, 4, 2

• 0, 3, -1, 4, 5, 3

Thus, we return 4.

Example 3:

Input: differences = 4,-7,2, lower = 3, upper = 6

Output: 0

Explanation: There are no possible hidden sequences. Thus, we return 0.

Constraints:

• n == differences.length

• <code>1 <= n <= 10⁵</code>

• <code>-10⁵<= differencesi<= 10⁵</code>

• <code>-10⁵<= lower <= upper <= 10⁵</code>