尼科马修斯定理(第k组奇数个正数的总和)

升序为1、3、5、7、9、11、13、15、17、19，…的正奇数。并分为(1)，(3、5)，(7、9、11)，(13、15、17、19)，…。等等。
因此，第一组是(1)，第二组是(3，5)，第三组是(7，9，11)，依此类推，通常，第k^个组包含该序列的后k个元素。
给定k ，求出第k^个组的总和。
例子 ：

Input : k = 3
Output : 27
3rd group is (7, 9, 11) and the sum is 27.

Input : k = 1
Output : 1

方法1：O(n)
这个想法是找到第k个组的第一个元素。
例如：

• 第一组的第一个元素为1，即第一个奇数。
• 第二组的第一个元素是3，这是第二个奇数。
• 第三组的第一个元素是7，这是第四个奇数。
• 第四组的第一个元素是13，这是第七个奇数。
• 等等。

通常，第k个组的第一个元素是第n个奇数，其中
n =(1 + 2 + 3 +……+(k – 1))+ 1。
现在，我们知道，第n个奇数是2n –1。这使我们成为第k个组的第一个元素。我们还知道该组中有k个元素，因此我们可以通过简单地从2n – 1向上计数两次，k次并将它们全部相加来找到它们的总和。这为我们提供了线性时间解决方案。

C++

// CPP program to find sum of k-th group of
// positive odd integers.
#include 
using namespace std;
 
// Return the sum of k-th group of positive
// odd integers.
int kthgroupsum(int k)
{
    // Finding first element of kth group.
    int cur = (k * (k - 1)) + 1;
    int sum = 0;
 
    // Finding the sum.
    while (k--) {
        sum += cur;
        cur += 2;
    }
 
    return sum;
}
 
// Driver code
int main()
{
    int k = 3;
    cout << kthgroupsum(k) << endl;
    return 0;
}

Java

// Java code to find sum of k-th group
// of positive odd integers.
import java.util.Arrays;
import java.util.Collections;
 
class GFG
{
    // Return the sum of k-th group
    // of positive odd integers.
    public static int kthgroupsum(int k)
    {
        // Finding first element of kth group.
        int cur = (k * (k - 1)) + 1;
        int sum = 0;
 
        // Finding the sum.
        while (k-- > 0)
        {
            sum += cur;
            cur += 2;
        }
 
        return sum;
    }
 
    // Driver program to test above methods
    public static void main(String[] args)
    {
        int k = 3;
        System.out.print(kthgroupsum(k));
    }
}
 
// This code is contributed by Chhavi

Python3

# Python3 code to find sum of k-th group
# of positive odd integers.
 
# Return the sum of k-th group of
# positive odd integers.
def kthgroupsum( k ):
     
    # Finding first element of kth group.
    cur = int((k * (k - 1)) + 1)
    sum = 0
     
    # Finding the sum.
    while k:
        sum += cur
        cur += 2
        k=k-1
     
    return sum
     
# Driver code
k = 3
print(kthgroupsum(k))
 
# This code is contributed by "Sharad_Bhardwaj".

C#

// C# code to find sum of k-th group
// of positive odd integers.
using System;
 
class GFG {
     
    // Return the sum of k-th group
    // of positive odd integers.
    public static int kthgroupsum(int k)
    {
         
        // Finding first element of kth group.
        int cur = (k * (k - 1)) + 1;
        int sum = 0;
 
        // Finding the sum.
        while (k-- > 0) {
            sum += cur;
            cur += 2;
        }
 
        return sum;
    }
 
    // Driver program to test above methods
    public static void Main()
    {
        int k = 3;
         
        Console.WriteLine(kthgroupsum(k));
    }
}
 
// This code is contributed by vt_m.

Javascript

// javascript program to find
// sum of k-th group of
// positive odd integers.
 
// Return the sum of
// k-th group of positive
// odd integers.
function kthgroupsum(k)
{
     
    // Finding first element
    // of kth group.
    let cur = (k * (k - 1)) + 1;
    let sum = 0;
 
    // Finding the sum.
    while (k--)
    {
        sum += cur;
        cur += 2;
    }
    return sum;
}
 
// Driver code
let k = 3;
document.write(kthgroupsum(k));
 
// This code is contributed by _saurabh_jaiswal.

输出 ：

27

方法2：O(1)
一个棘手的解决方案是找到k ³ 。可以很容易地观察到，第k组的总和为k ³ 。
以下是此方法的实现：

C++

// Efficient CPP program to find sum of k-th
// group of positive odd integers.
#include 
using namespace std;
 
// Return the sum of kth group of positive
// odd integer.
int kthgroupsum(int k)
{
    return k * k * k;
}
 
// Driven Program
int main()
{
    int k = 3;
    cout << kthgroupsum(k) << endl;
    return 0;
}

Java

// Efficient Java code to find sum of k-th
// group of positive odd integers.
import java.util.Arrays;
import java.util.Collections;
 
class GFG
{
    // Return the sum of kth group of
    // positive odd integer.
    public static int kthgroupsum(int k)
    {
        return k * k * k;
    }
 
    // Driver program to test above methods
    public static void main(String[] args)
    {
        int k = 3;
        System.out.print( kthgroupsum(k));
    }
}
 
// This code is contributed by Chhavi

Python3

# Efficient Python code to find sum of 
# k-th group of positive odd integers.
 
# Return the sum of kth group of positive
# odd integer.
def kthgroupsum( k ):
    return k * k * k
 
# Driver code
k = 3
print(kthgroupsum(k))
 
# This code is contributed by "Sharad_Bhardwaj".

C＃

// Efficient C# code to find sum of k-th
// group of positive odd integers.
using System;
 
class GFG {
     
    // Return the sum of kth group of
    // positive odd integer.
    public static int kthgroupsum(int k)
    {
        return k * k * k;
    }
 
    // Driver program to test above methods
    public static void Main()
    {
        int k = 3;
         
        Console.WriteLine(kthgroupsum(k));
    }
}
 
// This code is contributed by vt_m.

Java脚本

// Efficient Javascript program to
// find sum of k-th group of
// positive odd integers.
 
 
// Return the sum of kth group
// of positive odd integer.
function kthgroupsum(k)
{
    return k * k * k;
}
 
// Driven Code
let k = 3;
document.write(kthgroupsum(k));
 
// This code is contributed by _saurabh_jaiswal.

输出 ：

27