ABACABA

ABACABA, also known as the ABACABA sequence or pattern, is a self-similar, palindromic string generated by a simple recursive rule. It is defined by starting with the single letter A and, at each step, inserting the next letter of the alphabet in the center and duplicating the existing string on both sides. Formally, let S0 = "A". For n ≥ 1, Sn = Sn-1 + Ln + Sn-1, where Ln is the nth letter of the alphabet (B for n = 1, C for n = 2, etc.). The first few stages are: A; ABA; ABACABA; ABACABADABACABA; and so on.

A standard property of the ABACABA sequence is that every stage Sn is a palindrome. The length grows according to |Sn| = 2|Sn-1| + 1, with |S0| = 1, giving |Sn| = 2^{n+1} − 1. The center character at stage n is the nth letter of the alphabet. When extended with the usual alphabet, the sequence can be continued by using additional symbols or an expanded alphabet beyond Z, or by stopping at a chosen stage.

Variants and extensions of ABACABA often appear in discussions of recursion and self-similarity. The pattern serves as a simple, tangible example of how complex structures can emerge from a single rule and a growing central element. It is commonly used in recreational mathematics and computer science to illustrate ideas related to fractals, fixed points of rewriting systems, and palindrome properties.