About Palindrome II:

This sequence was first described by Clark Kimberling. It starts like this: 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0...

While an infinite integer sequence obviously can't be a palindrome, certain sections of this sequence closely resemble a palindrome. The sequence is actually based on the number of bit changes in the binary representation of the nth term. For instance, if n=8, then its binary representation is 1000, and looking from left to right, there is one bit change (from 1 to 0 between the leftmost and second-leftmost digits. So the 8th term in the sequence is 1. (For the first term in the sequence, n=1.)

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Reference: N. J. A. Sloane, editor (2002), The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences. Sequence A037834.