Minimum Garden Perimeter to Collect Enough Apples LeetCode solution in Python

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class Solution:
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    def minimumPerimeter(self, nap: int) -> int:
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        #         here for n = 2 , there are two series :
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        #         (1) Diagnal points for n=3 , diagnal apples = 2*n = 6
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        #         (2) there is series =   2,3,3 =  2+ (sigma(3)-sigma(2))*2
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        #         how to solve:
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        #             here 3  =  sigma(n+(n-1))-sigma(n) = sigma(2*n-1)-sigma(n) =  0.5*2n*(2n-1)-0.5*n*n-1
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        #         (3) so our final 2,3,3 =   3*2+2 =   (0.5*2n*(2n-1)-0.5*n*n-1)*2+n
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        #         (4) so final 2,3,3 =  3*n*n - 2*n
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        #         (5) we have 4 times repitation of (2,3,3)  = 4*(2,3,3)  =   4*(3*n*n - 2*n)   =  12*n*n - 8*n
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        #         (6) we have 4 diagnal points so their sum(4 diagnal)  = 4*(2*n)
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        #         (7)  so final sum(total) = 4 diagnal sum + 4(2,3,3)    =   4(2*n)   + 12*n*n - 8*n    =  12*n*n
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        #         so at nth distance we have total  12*n*n  apples at the circumfrance
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        #         so net sum =  sigma(12*n*n)  = 2*n*(n+1)*(2*n+1)
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        n = 1
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        val = 2 * n * (n + 1) * (2 * n + 1)
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        while val < nap:
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            n += 1
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            val = val = 2 * n * (n + 1) * (2 * n + 1)
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        return n * 8