Three Cubes And A Square

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

\[ 0^3 + 1^3 + 2^3 = 3^2,\quad 1^3 + 2^3 + 3^3 = 6^2 . \]

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

What is \(a+b+c?\)

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