HMMT 2002 T7

Let $T_{L}=\sum_{n=1}^{L}\left\lfloor n^{3} / 9\right\rfloor$ for positive integers $L$ . Determine all $L$ for which $T_{L}$ is a square number.

Dataset text NuminaMath olympiads. Quoted with source.

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L \in {1, 2}

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