FIBONACCI-TYPE SEQUENCES: NOVEL IDENTITIES, FAST COMPUTATION, AND COMBINATORIAL APPLICATIONS

Abstract

This paper presents a thorough investigation into generalized Fibonacci sequences defined by the second-order linear recurrence

with arbitrary initial conditions ­[9]. Utilizing matrix formulation and diagonalization ­[10], we derive efficient  algorithms for sequence computation and establish Binet-like closed-form solutions ­[11]. Novel identities, including the product-difference identity [12], are rigorously proven through matrix determinant techniques [10], induction, and generating functions [14]. We analyze convergence properties of generalized sequences, identifying spectral conditions ensuring ratio limits tied to dominant eigenvalues [15]. The classical Fibonacci sequence’s combinatorial richness is revisited via Zeckendorf’s theorem [16], establishing unique non-consecutive sum representations and their applications to optimal Fibonacci coding [17] with near-entropic average code length. Advanced number-theoretic considerations include Pisano periods [18], offering bounds on periodicity modulo integers through algebraic and finite field techniques [19].

Finally, we demonstrate fast computational schemes based on doubling formulas [20] and matrix squaring to enable logarithmic complexity calculations of Fibonacci numbers [21].