Zeckendorfin

Zeckendorfin is a fictional mathematical construct used to illustrate Zeckendorf representations of integers within the Fibonacci system. It is typically described as a finite directed acyclic graph or automaton whose states encode partial sums built from Fibonacci numbers and whose transitions enforce the nonconsecutive index rule that underlies Zeckendorf's theorem. The term is used primarily in teaching and thought experiments to connect number representations with automata theory.

Formally, take the Fibonacci sequence with F1 = 1, F2 = 2, and Fk = F{k-1} + F{k-2} for k ≥ 3. A Zeckendorfin path represents a Zeckendorf representation of an integer n as a set of indices S such that n = sum_{i in S} F_i, with no two consecutive indices in S. The Zeckendorfin graph contains a start state and terminal states; edges labeled by F_i indicate that including F_i advances the representation, and the nonconsecutivity constraint is implemented by forbidding transitions that would choose adjacent indices.

A key property is that Zeckendorf representations are unique, and a well-constructed Zeckendorfin graph can realize exactly one accepting path for each representable integer. The structure can be used to simulate conversion between integers and their Zeckendorf forms and to study counting questions related to constrained sums of Fibonacci numbers.

Variants of the Zeckendorfin idea include graphs built from generalized Fibonacci sequences or from alternative constraints on index adjacency (for example, allowing at most k consecutive indices). Such variants are discussed in hypothetical teaching resources and illustrate the broader connections between number representations, automata, and combinatorics.

Because Zeckendorfin is a fictional construct, it does not appear in standard mathematical literature, but the concept echoes the well-established Zeckendorf's theorem and related Fibonacci coding schemes. It is sometimes used to clarify ideas in introductory courses on discrete math and automata theory.

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