Is there an explicit or recursive formula for the sequence 1,-3, 5, -7, 9...

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Is there an explicit or recursive formula for the sequence 1,-3, 5, -7, 9...

A fifth grader could probably explain the rule in words, but I'm challenged in doing so mathematically. I noticed that the 'second differences' are always 4, possibly implying an underlying quadratic, but utterly baffled by the sign-flipping. Any ideas? Thank you. VJ

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Let $a_1=1$. $$a_n=(-1)^{n+1}\times(2n-1)$$ or $$a_n=(-1)^{n+1}(|a_{n-1}|+2)$$