# 413. Arithmetic Slices

A sequence of number is called arithmetic if it consists of at least three elements and if the difference between any two consecutive elements is the same.

For example, these are arithmetic sequence:

1, 3, 5, 7, 9 7, 7, 7, 7 3, -1, -5, -9

The following sequence is not arithmetic.

1, 1, 2, 5, 7

A zero-indexed array A consisting of N numbers is given. A slice of that array is any pair of integers (P, Q) such that 0 <= P < Q < N.

A slice (P, Q) of array A is called arithmetic if the sequence: A[P], A[p + 1], ..., A[Q - 1], A[Q] is arithmetic. In particular, this means that P + 1 < Q.

The function should return the number of arithmetic slices in the array A.

Example:

A = [1, 2, 3, 4]
return: 3, for 3 arithmetic slices in A: [1, 2, 3], [2, 3, 4] and [1, 2, 3, 4] itself.

# Solution

Approach 1: DP. We can save space by keeping just the previous item.

# Code (Python)

Approach 1:

class Solution:
    def numberOfArithmeticSlices(self, nums: List[int]) -> int:
        if len(nums) < 3:
            return 0

        diffs = [nums[i] - nums[i-1] for i in range(1, len(nums))]
        # num_slices[i] -- number of constant diffs that ends with diffs[i]
        num_slices = [0 for _ in range(len(diffs))]
        for i in range(1, len(diffs)):
            if diffs[i] == diffs[i-1]:
                num_slices[i] = 1 + num_slices[i-1]
        return sum(num_slices)
    
    def numberOfArithmeticSlices(self, nums):
        # save space using a "window"
        if len(nums) < 3:
            return 0
        diffs = (nums[i] - nums[i-1] for i in range(1, len(nums)))
        diff1, diff2 = float('inf'), next(diffs)
        slices1, slices2 = 0, 0
        num_slices = 0
        for new_diff in diffs:
            diff1, diff2 = diff2, new_diff
            if diff2 == diff1:
                slices1, slices2 = slices2, slices2 + 1
            else:
                slices1, slices2 = slices2, 0
            num_slices += slices2
        return num_slices

# Code (C++)

Approach 1:

Approach 2: