Representation of Natural Numbers as Sums of Generalised Fibonacci Numbers

Generalizing Zeckendorf's Theorem to f -decompositions

“…For every such recurrence relation there is a notion of "legal decomposition" with which all positive integers have a unique decomposition as a non-negative integer linear combination of terms from the sequence, and the distribution of the number of summands of integers in [G n , G n+1 ) converges to a Gaussian. There is an extensive literature for this subject; see [Al,BCCSW,Day,GT,Ha,Ho,Ke,Len,MW1,MW2] for results on uniqueness of decomposition, [DG,FGNPT,GTNP,KKMW,Lek,LamTh,MW1,St] for Gaussian behavior, and [BBGILMT] for recent work on the distribution of gaps between summands. An alternative definition of the Fibonacci sequence can be framed in terms of the Zeckendorf non-consecutive condition: The Fibonacci sequence (beginning F 1 = 1, F 2 = 2) is the unique increasing sequence of natural numbers such that every positive integer can be written uniquely as a sum of non-consecutive terms from the sequence.…”

Sum-distinct sequences and Fibonacci numbers

“…The concept of that type of generalized Fibonacci numbers was introduced and investigated first by Daykin in [3]; our definition corresponds to (t, t)-Fibonacci numbers there. One of their useful properties is formulated in the following proposition.…”

Unified algebraic synthesis of generalized Fibonacci Switched Capacitor Converters

“…SIGNED GENERALIZED FIBONACCI (SGF) REPRESENTATION The proposed approach to synthesis of generalized Fibonacci SCC is based on the novel number system described in this section. The generalized (h, k)-th Fibonacci numbers [8]- [10] are defined for i ≥ 2 and h ≤ k ≤ h+1 as:…”