About Palindrome I:
This sequence was first described by Flajolet and Ramshaw. It starts like this: 0, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1...
Like many of the other sequences here, this sequence is self-similar; it contains infinitely many copies of itself and is invariant under scale. If you create a new integer sequence by using every third term of this one, you end up with exactly the same sequence!
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 "weight of balanced ternary representation of n." I must confess I have no idea what that means.
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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 A005812.