Sum is 2

If you run the contour walker for N=2, you will get a lot of hits for a sum of 2, for instance:

• 49^3-47^3-24^3
• 163^3-161^3-54^3
• 385^3-383^3-96^3
• 751^3-749^3-150^3
• 1297^3-1295^3-216^3
• 2059^3-2057^3-294^3
• ...
• 10841593393^3-10841593391^3-8901144^3
• ...

10,841,593,393 was the largest value found when I stopped the search.

Inspection reveals that these are all of the form:

        (6n^3+1)^3-(6n^3-1)^3-(6n^2)^3 = 2

The above list starts with n = 2. if we replace the ±1 in the above expression with ±a^3 we can get the more general:

(6n^3+a^3)^3-(6n^3-a^3)^3-(6n^2a)^3=2a^3.

We can also multiply all the terms by b, and thus generate an infinite number of sums that add up to any constant of the form 2*a^9*b^3.