偶数和奇数自然数的多维数据集的总和

我们知道，前n个自然数的立方和为=(n(n + 1)/ 2) ² 。

前n个偶数自然数的立方和
2 ³ + 4 ³ + 6 ³ +………+(2n) ³

Even Sum = 23 + 43 + 63 + .... + (2n)3
        if we multiply by 23 then 
        = 23 x (13 + 23 + 32 + .... + (n)3)
        = 23 + 43 + 63 + ......... + (2n)3
        =  23 (n(n+1)/2)2
        =  8(n(n+1))2/4
        =  2(n(n+1))2

例子 ：

Sum of cube of first 4 even numbers = 23 + 43 + 63 + 83 
 put n = 4     = 2(n(n+1))2
               = 2*(4*(4+1))2
               = 2(4*5)2
               = 2(20)2
               = 800
 8 + 64 + 256 + 512 = 800

前n个偶数的立方和的程序

前n个奇数自然数的立方和
我们需要计算1 ³ + 3 ³ + 5 ³ +…。 +(2n-1) ³

OddSum   = (Sum of cubes of all 2n numbers) - (Sum of cubes of first n even numbers)
         =  (2n(2n+1)/2)2 - 2(n(n+1))2 
         =  n2(2n+1)2 - 2* n2(n+1)2
         =  n2[(2n+1)2 - 2*(n+1)2]
         =  n2[4n2 + 1 + 4n - 2n2 - 2 - 4n]
         =  n2(2n2 - 1)

例子 ：

Sum of cube of first 4 odd numbers = 13 + 33 + 53 + 73 
 put n = 4     = n2(2n2 - 1)
               = 42(2*(4)2 - 1)
               = 16(32-1)
               =  496
 1 + 27 + 125 + 343 = 496

前n个奇数的立方和的程序