Three Cubes And A Square

The triplets of consecutive integers (0,1,2)(0,1,2) and (1,2,3)(1,2,3) have the property that the sum of their cubes is a perfect square:

03+13+23=32,13+23+33=62. 0^3 + 1^3 + 2^3 = 3^2,\quad 1^3 + 2^3 + 3^3 = 6^2 .

Find the next smallest triplet of consecutive integers (a,b,c)(a, b, c) that has this property.

What is a+b+c?a+b+c?

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