numeronlike

Numeronlike is a term used in theoretical and recreational discussions of self-generative numeric patterns. It describes a broad class of objects—most often numeric sequences or digit-structured strings—that are produced from a fixed, local rule that reads the current term and outputs the next. The term is not tied to a single formal theory but serves as a linguistic label for patterns that exhibit self-reference and iterative growth.

Formal concept: Let base b be a positive integer, and let a_0 be an initial term. A numeronlike sequence in base b is a sequence (a_n) where each a_{n+1} is obtained from a_n by applying a fixed operation O that depends only on a_n (and possibly on b). The operation typically analyzes the digits of a_n (in base b), performs a simple transformation, and concatenates results to form a_{n+1}. The same rule is used at every step.

Examples: The classic look-and-say sequence in base 10 is a canonical numeronlike process: a_0 = "1", and a_{n+1} is the description of a_n's digits. Variants include digit-counting schemes that emit run-lengths and symbols, or rules that map digit counts to new digits. Static analogues such as self-descriptive numbers are related in spirit but do not form a progression.

Notes: The base, initial term, and exact rule heavily influence growth, structure, and complexity. In general, numeronlike patterns illustrate themes of self-reference, information growth, and emergent structure and are discussed mainly in recreational mathematics and theoretical discussions of algorithms and automata.