In the sequence 1, 0, 1, 0, 1, 0, 3, 5... each member after the sixth one equals the units' digit of the sum of the six preceding numbers of the sequence.

Prove that the subsequence 0, 1, 0, 1, 0, 1, will never occur.

I just calculated the sequence which starts with the target subsequence
101010. (I used excel, my tool of preference). Interesting
enough, it also repeated every 1456 numbers. It did not include
010101, so this is another way to show that one will not generate the
other.

I sure hope somebody comes up with something elegant.