The numbers which are subtracted from the Root form a series
If one looks at the sum, the number of terms relates to the cube root of the sum itself.
1:~ 1
2:~ 1 + 7
3:~ 1 + 7 + 19
4:~ 1 + 7 + 19 + 37
5:~ 1 + 7 + 19 + 37 + 61
.
.
n:~ 1 + 7 + 19 + 37 + 61 + ....... +
Notice also that these terms can be written as
n:~ 1 + (1+6) + (1+18) + (1+36) + (1+60) + .... +
= 1 + (1 + 6) + (1 +6 + 12) + (1+6+12+18) + (1+6+12+18+24) + ....
= 1 + (1 + A) + (1 + B) + (1 + C) + (1 + D) +
Where
A = 6
B = 6 + 12
C = 6 + 12 + 18
D = 6 + 12 + 18 + 24
1 occurs n times
6 = 6.1 occurs n-1 times
12 = 6.2 occurs n-2 times
18 = 6.3 occurs n-3 times
24 = 6.4 occurs n-4 times
The term which occurs 1 times only will be 6.(n-1)
Thus the sum becomes
Therefore: