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, 97, 7, 7, 73, -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.

从头开始扫描，每当能够形成arithmetic sequence的时候都把尽量多的元素加入进来，例如给定1,3,5,7,9

扫描到1,3,5已经能够构成arithmetic sequence了，这时候尽可能地继续扩展最后找到1,3,5,7,9 

为什么尽可能地扩展呢？因为例如1,3,5,7,9是arithmetic sequence，除了这个序列本身之外，它的包含4个元素的子集、包含3个元素的子集也是

arithmetic sequence，这样只需要数一下这个序列有多少个包含>=3个元素的子集就可以了，这样的子集有(n-2)(n+1-2)/2 个

public class Solution {    public int numberOfArithmeticSlices(int[] A){int len=A.length;if(len<3)return 0;int sum=0;for(int i=0;i<len-2;){int diff=A[i+1]-A[i];int start=i;i++;while(i+1<len&&A[i+1]-A[i]==diff)i++;int count=i+1-start;if(count>=3)sum+=(count-2)*(count-1)/2;}return sum;}}

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dp法：

https://discuss.leetcode.com/topic/62992/3ms-c-standard-dp-solution-with-very-detailed-explanation/2

// dp[i] means the number of arithmetic slices ending with A[i]

  // if A[i-2], A[i-1], A[i] are arithmetic, then the number of arithmetic slices ending with A[i] (dp[i])            // equals to:            //      the number of arithmetic slices ending with A[i-1] (dp[i-1], all these arithmetic slices appending A[i] are also arithmetic)            //      +            //      A[i-2], A[i-1], A[i] (a brand new arithmetic slice)            // it is how dp[i] = dp[i-1] + 1 comes

public static int numberOfArithmeticSlices(int[] A){int len=A.length;if(len<3)return 0;int[] dp=new int[len];int ret=0;for(int i=2;i<len;i++){if(A[i]-A[i-1]==A[i-1]-A[i-2])dp[i]=dp[i-1]+1;ret+=dp[i];}return ret;}