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n^2 + (n+2)^2 = (n+4)^2 - 15
2n^2 + 4n +4 = n^2 + 8n +16 - 15
n^2 - 4n + 3 = 0
(n - 1)(n - 3) = 0
n = 1 or n = 3

1 and 3 both fulfill the condition, because

1^2 + 3^2 = 5^2 - 15
i.e. 1 + 9 = 25 - 15 = 10

and

3^2 + 5^2 = 7^2 - 15
i.e. 9 + 25 = 49 - 15 = 34

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