LeetCode.441 Arranging Coins （经典数列求和应用）

题目：

You have a total of n coins that you want to form in a staircase shape, where every k-th row must have exactly k coins.

Given n, find the total number of full staircase rows that can be formed.

n is a non-negative integer and fits within the range of a 32-bit signed integer.

Example 1:

n = 5The coins can form the following rows:¤¤ ¤¤ ¤Because the 3rd row is incomplete, we return 2.

Example 2:

n = 8The coins can form the following rows:¤¤ ¤¤ ¤ ¤¤ ¤Because the 4th row is incomplete, we return 3.

分析：

class Solution {    public int arrangeCoins(int n) {        //给定总的砖数，具体第k台阶满足用k块砖，求具体满足了多少层台阶        //思路：类似数列求和,求满足k-1满足小于n，k满足n的数        //暴力解法：TLM        // if(n==0) return 0;        // for(int i=1;i<=Math.sqrt(n);i++){        //     if(((1+i)*i/2<=n)&&((i+1+1)*(i+1)/2>n)){        //         return i;        //     }        // }        // return 0;                //巧妙解法        //         `(x * ( x + 1)) / 2 <= n`        // Using quadratic formula, `x` is evaluated to be,        //利用二元一次方程求解 x=（-b+_sqrt(b^2-4ac)）/2a        // `x = 1 / 2 * (-sqrt(8 * n + 1)-1)` (Inapplicable) or `x = 1 / 2 * (sqrt(8 * n + 1)-1)`        // Negative root is ignored and positive root is used instead. Note that 8.0 * n is very important because it will cause Java to implicitly autoboxed the intermediate result into double data type. The code will not work if it is simply 8 * n.                return (int)((Math.sqrt(1+8.0*n)-1)/2);    }}