Fibonaccinévre

Fibonaccinévre is a fictitious mathematical construct used in recreational mathematics and puzzle design. The name blends Fibonacci with névre, a neologism coined to evoke neural-like perturbations in a standard recurrence.

One common formalization defines a Fibonaccinévre sequence {F(n)} by F(0)=0, F(1)=1, and for n≥2, F(n) = F(n-1) + F(n-2) + b(n), where b(n) is a fixed binary control sequence with period p. The term b(n) introduces a localized perturbation to the classic Fibonacci recursion.

Example: With b(n) = 1 if n is divisible by 3, and b(n) = 0 otherwise, the first terms are: 0, 1, 1, 3, 4, 7, 12, 19, 31, 51, 82, ...

Variants: Other choices of b(n) (different periods or nonperiodic rules) yield diverse sequences, all sharing the underlying Fibonacci backbone. Researchers in puzzle communities study asymptotic behavior, growth rate, and ratio of successive terms, which often approaches the golden ratio with fluctuations depending on b(n).

Applications and reception: Fibonaccinévre is primarily used for teaching and recreational exploration of recurrences, illustrating how small, periodic perturbations affect well-known sequences. It has appeared in online articles, coding challenges, and classroom demonstrations as a neutral example of perturbations in linear recurrences.

See also: Fibonacci sequence, linear recurrence relations, perturbations in recurrences, Beatty sequences.