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 Alternating sextet (Posted on 2007-03-30)
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.

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 Improvement after 10 years Comment 6 of 6 |
It is slightly more elegant to consider the series modulus 5.  It then has a cycle length of only 208 (instead of 1456) and 010101 does not appear, which proves that it will never appear.

It also proves that 63 other sequences will never occur, all of which equal 0,1,0,1,0,1 mod 5.  For instance, 5,6,5,6,5,6 and 5,6,0,1,5,1 will never occur.

Note:  This approach only works because 5 divides 10.  Considering the series modulus 3 would prove nothing.

 Posted by Steve Herman on 2017-05-05 18:58:46

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