What is Zeckendorf’s Theorem?
نویسنده
چکیده
While Fibonacci numbers can quite easily be classified as a complete sequence, they have the unusual property that a particular explicitly defined subset of such sequences is complete and also gives unique representations of natural numbers. This can be extended to all integers via the ”nega-Fibonacci” numbers. Further, an unusual binary operation on the natural numbers may be defined and proven to be associative.
منابع مشابه
Zeckendorf's Theorem and Fibonacci Coding for Modules
Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers. This theorem induces a binary numeration system for the positive integers known as Fibonacci coding. Fibonacci code is a variable-length prefix code that is robust against insertion and deletion errors and is useful in data transmission and data compression. In this pape...
متن کاملA GENERALIZATION OF ZECKENDORF’S THEOREM VIA CIRCUMSCRIBED m-GONS
Zeckendorf’s theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy Fn = Fn−1 + Fn−2 for n ≥ 3, F1 = 1 and F2 = 2. The distribution of the number of summands in such decomposition converges to a Gaussian, the gaps between summands converges to geometric decay, and the distribution of the longest gap...
متن کاملA Constant Optimization of the Binary Indexed Tree Query Operation
There are several data structures which can calculate the prefix sums of an array efficiently, while handling point updates on the array, such as Segment Trees and Binary Indexed Trees (BIT). Both these data structures can handle the these two operations (query and update) in O(log n) time. In this paper, we present a data structure similar to the BIT, but with an even smaller constant. To do t...
متن کاملExplicit Constructions of Memoryless Crosstalk Avoidance Codes via C-transform
One of the main problems in deep sub-micron designs of high speed buses is the propagation delay due to the crosstalk effect. To alleviate the crosstalk effect, there are several types of crosstalk avoidance codes proposed in the literature. In this technical brief, we develop explicit constructions of two types of memoryless crosstalk avoidance codes: forbidden overlap codes (FOCs) and forbidd...
متن کاملA New Approach to File Encryption and Compression
In this paper, a new compression/encryption scheme is presented. The basic idea of this scheme is to represent a monotonically increasing sequence of integers with a single integer such that the original sequence can be restored perfectly. First, we recognize the binary bit patterns from the input file using Lempel-Ziv incremental parsing algorithm. Then a second pass through the file revises t...
متن کامل