$n=1$ and $n=3$ are not solutions. Here is a construction for $n\ne 1,3$: -For $n=2$: $01$ -For $n=4k$: $01100110...0110$ ($k$ groups of $0110$) -For $n=4k+1$: $01100110...011001010$ ($k-1$ groups of $0110$ and $01010$ at the end) -For $n=4k+2$: $01100110...011001$ ($k$ groups of $0110$ and $01$ at the end) -For $n=4k+3$: $01100110...0110001$ ($k$ groups of $0110$ and $001$ at the end) (I write $0$ for a lamp which is on and $1$ for a lamp which is off)