ABACABADABACABA

ABACABADABACABA is a self-similar, palindromic string derived from the ABACABA pattern, a classic recursive construction. It represents the fourth-level instance in a family of strings formed by placing a new central letter around a copy of the previous string.

Construction and example: Let S1 be "A". For n ≥ 1, define S_{n+1} = S_n + L_n + S_n, where L_n is the nth letter of the alphabet (A, B, C, D, ...). This yields S2 = "ABA", S3 = "ABACABA", and S4 = "ABACABADABACABA". The item ABACABADABACABA is precisely S4.

Properties: The string is a palindrome, with the center letter D. It uses only the letters A through D. Its length is 15 characters, since |S4| = 2|S3| + 1 = 15. The construction is inherently recursive, and the pattern can be extended to higher levels by continuing with the next alphabet letter (E, and so on).

Usage and context: The ABACABA family is frequently cited in discussions of recursion, combinatorics on words, and puzzle design to illustrate how simple rules can generate complex, self-similar structures. ABACABADABACABA serves as a concrete, compact example due to its symmetry and straightforward construction.

See also: ABACABA pattern, recursive sequence, palindrome.