3700. Number of ZigZag Arrays II
You are given three integers n, l, and r.
A ZigZag array of length n is defined as follows:
- Each element lies in the range
[l, r]. - No two adjacent elements are equal.
- No three consecutive elements form a strictly increasing or strictly decreasing sequence.
Return the total number of valid ZigZag arrays.
Since the answer may be large, return it modulo 109 + 7.
A sequence is said to be strictly increasing if each element is strictly greater than its previous one (if exists).
A sequence is said to be strictly decreasing if each element is strictly smaller than its previous one (if exists).
Example 1:
Input: n = 3, l = 4, r = 5
Output: 2
Explanation:
There are only 2 valid ZigZag arrays of length n = 3 using values in the range [4, 5]:
[4, 5, 4][5, 4, 5]
Example 2:
Input: n = 3, l = 1, r = 3
Output: 10
Explanation:
There are 10 valid ZigZag arrays of length n = 3 using values in the range [1, 3]:
[1, 2, 1],[1, 3, 1],[1, 3, 2][2, 1, 2],[2, 1, 3],[2, 3, 1],[2, 3, 2][3, 1, 2],[3, 1, 3],[3, 2, 3]
All arrays meet the ZigZag conditions.
Constraints:
3 <= n <= 1091 <= l < r <= 75
Hints
Hint 1
Hint 2
2*m where m = r - l + 1: first m entries = "next compare = down" for values, next m = "next compare = up".Hint 3
T (size 2*m × 2*m): from an up,x state go to down,y for every y > x, and from down,x go to up,y for every y < x.Hint 4
T^(n-1), apply it to the initial vector (ones in the block for starting up and separately for starting down), sum final entries, and add both results (for n=1 return m).