Generalization of Binet's formula for Fibonacci-type numeric sequences through the use of arithmetic pseudo-operators

This paper presents an innovative approach to the study of recurrent sequences by introducing the concept of arithmetic pseudo-operators. Unlike conventional operators, these pseudo-operators are pure complex numbers with specific structural properties, allowing for unprecedented operational reformulations. Represented by the symbols $+$'', $\mathrm{/}$‘’ (slash), $\setminus$'' (aslash), $\mathrm{\perp }$’‘, $\top$'', and $⊣$’‘, these operators correspond to rotations in the unit circle of the complex plane and generalize fundamental operations, as seen in the identities $1 / 1 \setminus 1 = 0$'' and $1\mathrm{\perp }1\mathrm{\top }1⊣1=0$’‘, which exhibit behavior analogous to conventional subtraction ``$1-1=0$’'. Based on this structure, we reformulate Binet’s equations for the Fibonacci and Tribonacci sequences and outline the path for their generalization to the Tetranacci sequence \cite{koshy}. This new perspective not only enhances the understanding of higher-order recurrences but also suggests potential applications in discrete mathematics and computational algebra, expanding the scope of classical algebraic operations.