count-the-hidden-sequences

Count the Hidden Sequences

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

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

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

	<ul>
		<li><code>[3, 4, 1, 5]</code> and <code>[4, 5, 2, 6]</code> are possible hidden sequences.</li>
		<li><code>[5, 6, 3, 7]</code> is not possible since it contains an element greater than <code>6</code>.</li>
		<li><code>[1, 2, 3, 4]</code> is not possible since the differences are not correct.</li>
	</ul>
	</li>
</ul>

Return the number of possible hidden sequences there are. If there are no possible sequences, return <code>0</code>.

&nbsp;
Example 1:

<pre>
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.
</pre>

Example 2:

<pre>
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.
</pre>

Example 3:

<pre>
Input: differences = [4,-7,2], lower = 3, upper = 6
Output: 0
Explanation: There are no possible hidden sequences. Thus, we return 0.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>n == differences.length</code></li>
	<li><code>1 &lt;= n &lt;= 105</code></li>
	<li><code>-105 &lt;= differences[i] &lt;= 105</code></li>
	<li><code>-105 &lt;= lower &lt;= upper &lt;= 105</code></li>
</ul>

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