Fibonaccitype

Fibonaccitype is a term used in mathematics to describe a class of sequences and related structures whose growth is governed by a Fibonacci-like recurrence. In its simplest form a fibonaccitype is a sequence {f(n)} defined by f(n) = f(n-1) + f(n-2) for n ≥ 2 with chosen initial values f(0) and f(1). Such sequences exhibit exponential growth characteristic of the Fibonacci family, and many combinatorial objects counted by fibonaccitypes share this growth pattern.

Origin and scope: The term is used in certain theoretical and expository contexts to unify counting problems where a Fibonacci-type recurrence arises naturally. Classic examples include tilings with squares and dominoes, binary strings with no consecutive ones, and restricted-part compositions. A fibonaccitype is thus not a single sequence but a class of structures whose counts can be described by a Fibonacci-type recurrence.

Mathematical properties: Solutions to f(n) = f(n-1) + f(n-2) admit a closed form via a Binet-like formula, and grow asymptotically as a constant multiple of φ^n, where φ is the golden ratio ≈ 1.618… The ratio f(n)/f(n-1) converges to φ as n increases. Zeckendorf's theorem, which expresses every positive integer as a sum of nonconsecutive Fibonacci numbers, highlights the combinatorial role of fibonaccitype counts in representations.

Variants and extensions: Generalizations include sequences with different initial values, vector-valued fibonaccitypes, and higher-order recurrences that preserve Fibonacci-like growth patterns. Fibonaccitypes also relate to constructs such as Fibonacci tilings and the Fibonacci word, where the counting sequences are Fibonacci numbers.

See also Fibonacci numbers, Zeckendorf representation, Fibonacci tiling, Fibonacci word.