Derivatives of Binary Sequences
Melvyn B. Nathanson · SIAM Journal on Applied Mathematics · 1971
A binary sequence is a sequence of 0’s and l’s. If $G = ( {g_i } )_{i = 0}^\infty $ is a binary sequence, the derivative of G, denoted $D( G )$, is the binary sequence $D( G ) = ( {g_i + g_{i + 1} } )_{i = 0}^\infty $, where the addition is modulo $2( {0 + 0 = 1 + 1 = 0,0 + 1 = 1 + 0 = 1} )$. By taking successive derivatives, we generate the sequence of derivatives of G, denoted $\mathcal{D}( G )$, $\mathcal{D}( G ) = ( {D^{( n )} ( G )} )_{n = 0}^\infty $. Extending recent results of T. Goka [1], we prove: The binary sequence G is eventually periodic if and only if its sequence of derivatives $\mathcal{D}( G )$ is eventually periodic.