DK-GEORGMOHR 2021 Problem 3

Georg investigates which integers are expressible in the form $$\pm 1^2 \pm 2^2 \pm 3^2 \pm \dots \pm n^2.$$ For example, the number $3$ can be expressed as $-1^2 + 2^2$ , and the number $-13$ can be expressed as $+1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2$ . Are all integers expressible in this form?