OEIS A000034: The deceptively simple alternating 1-2 sequence

We explore A000034, the 1,2,1,2,... sequence defined by a(n) = 1 + n mod 2, and its surprising connections. Beyond its simple pattern, it links to the base-3 digital root of n+1, a simple continued fraction for a value related to sqrt(3), and a distinctive Hankel transform of 1, -3, 0, 0, 0,... It is lexicographically the earliest sequence with a certain polynomial-fitting property and it encodes maximal anti-chains in power-set lattices. The sequence admits many closed forms and relationships: a(n) = 2^{(1-(-1)^n)/2}, a(n) = gcd(n-1, n+1), and a(n) = a01704(n)/3, among others, illustrating how a tiny pattern can reveal deep connections across number theory, combinatorics, and beyond.

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