Explicit definition of the binary reflected Gray codes

It is shown that for 1 ⩽ j ⩽ n and 1 ⩽ k ⩽ 2″, the jth letter of the kth word of the binary reflected Gray code of length n is equal to the parity of the binomial coefficient 2n−2n−j−1C[2n−2n−j−1−k/2] modulo 2. Also it is shown how this observation and the usual iterative definition of the binary reflected Gray codes are revealed in a modified version of Sierpinskiʹs gasket (Pascalʹs triangle modulo 2).