Again in mod 2, a companion sequence
oeis.org/A392906
[0]
[1, 0]
[1, 1, 0]
[0, 1, 0, 0]
[0, 0, 1, 0, 0]
[0, 0, 1, 0, 0, 0]
[0, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0]
[0, 0, 1, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 0, 1, 0, 1, 0, 0, 0]
[0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0]