Heron and His Triangle HDU - 6222（pell 大数）

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Heron and His Triangle

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 262144/262144 K (Java/Others)
Total Submission(s): 2543    Accepted Submission(s): 1060

Problem Description

A triangle is a Heron’s triangle if it satisfies that the side lengths of it are consecutive integers t−1, t, t+ 1 and thatits area is an integer. Now, for given n you need to find a Heron’s triangle associated with the smallest t bigger
than or equal to n.

 

Input

The input contains multiple test cases. The first line of a multiple input is an integer T (1 ≤ T ≤ 30000) followedby T lines. Each line contains an integer N (1 ≤ N ≤ 10^30).

 

Output

For each test case, output the smallest t in a line. If the Heron’s triangle required does not exist, output -1.

 

Sample Input

4
1
2
3
4

 

Sample Output

4
4
4
4

 

Source

2017ACM/ICPC亚洲区沈阳站-重现赛（感谢东北大学）

 

Recommend

jiangzijing2015

遇到求三角形面积且已知三边长 那就用海伦公式

Heron and His Triangle

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 262144/262144 K (Java/Others)
Total Submission(s): 2543    Accepted Submission(s): 1060

Problem Description

A triangle is a Heron’s triangle if it satisfies that the side lengths of it are consecutive integers t−1, t, t+ 1 and thatits area is an integer. Now, for given n you need to find a Heron’s triangle associated with the smallest t bigger
than or equal to n.

 

Input

The input contains multiple test cases. The first line of a multiple input is an integer T (1 ≤ T ≤ 30000) followedby T lines. Each line contains an integer N (1 ≤ N ≤ 10^30).

 

Output

For each test case, output the smallest t in a line. If the Heron’s triangle required does not exist, output -1.

 

Sample Input

4
1
2
3
4

 

Sample Output

4
4
4
4

 

Source

2017ACM/ICPC亚洲区沈阳站-重现赛（感谢东北大学）

 

Recommend

jiangzijing2015

首先根据题意设a=t-1，b=t，c=t+1，这样我们可由海伦凯勒公式得到

那么要想使得s为整数，就要使      等于3乘以一个平方数

所以我们设

于是得到               d=3

根据题意可知最小特解为x=2，y=1；

打个表之后发现这样一些数字4,14,52,194,724,2702....然后得出递推式子，F[n]=4*F[n-1]-F[n-2];由于n非常的大，所以矩阵快幂维护也不行。最后考虑这样的数字几乎增长比较快，那么范围内这样的数字就会比较少，不想用高精度的可以考虑用java大数了，用个list将所有n范围内的结果保存一下，最后直接查询就可以了。