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:
The following sequence is not arithmetic.
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.
- state: dp[i] means the number of arithmetic slices ends with A[i]
- function: dp[i] = dp[i - 1] + 1. A[i] 跟dp[i - 1]中的每一个方案都成再组成一个数列，但是增加了一个包含A[i]的长度为3的数列。
- result: sum of dp[i]
(n - 1) * (n - 2) / 2个。
一个长度为n的数列n (n >= 3), 其中有长度为3的子数列n - 2个, 长度为4的子数列n - 3个，以此类推，直到1. 这是一个等差数列。
按照等差数列求和公式：sum = (a1 + an) n / 2. 依本题代入 (1 + n - 2) (n - 2) / 2 = (n - 1) * (n - 2) / 2.