This paper will explore the relationship between the Fibonacci numbers and the Euclidean Algorithm in addition to generating a generalization of the Fibonacci Numbers. It will also look at the ratio of adjacent Fibonacci numbers and adjacent generalized Fibonacci numbers. Finally it will explore some fun applications and properties of the Fibonacci numbers.

To overcome many drawbacks in the current JPEG standard for sill image compression, a new standard, JPEG2000, is under development by International Standard Organization. The compact representation of integers is an important problem in areas such as data compression, especially where there is a nearly monotonic decrease in the likelihood of large integers. While describing many representation,...

The number of binary strings of length n containing no substrings consisting of r consecutive ones is examined and shown to be given in terms of a well known integer sequence namely, the r-Fibonacci sequence. In addition, difference equations for the number of zeros and the total number of runs within these binary strings are derived. Mathematics Subject Classification: Primary 11B39, Secondary...

1. A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 3 (1965):161-75. 2. A. F. Horadam. "Complex Fibonacci Numbers and Fibonacci Quaternions." Amer. Math. Monthly 70 (1963):289-91. 3. A. L. Iakin. "Generalized Quaternions with Quaternion Components." The Fibonacci Quarterly 15 (1977):35Q-52. 4. A. L. Iakin. "Generalized Quaternions of Higher...

The objective of this study was to develop a new optimal parallel algorithm for matrix multiplication which could run on a Fibonacci Hypercube structure. Most of the popular algorithms for parallel matrix multiplication can not run on Fibonacci Hypercube structure, therefore giving a method that can be run on all structures especially Fibonacci Hypercube structure is necessary for parallel matr...

We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1, 2, . . . , n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.

1 Morse code polynomials Morse code sequences are finite sequences of dots (•) and dashes (−). If a dot has length 1 and a dash has length 2 then the number of all such sequences of total length n−1 is the Fibonacci number Fn, which is defined as the sequence of numbers satisfying the recursion Fn = Fn−1 + Fn−2 with initial conditions F0 = 0 and F1 = 1. If a dash is assumed to have length j, j≥...

1. J. C. Butcher. "On a Conjecture Concerning a Set of Sequences Satisfying The Fibonacci Difference Equation/ The Fibonacci Quarterly 16 (1978):8183. 2. M. D. Hendy, "Stolarskys Distribution of Positive Integers." The Fibonacci Quarterly 16 (1978)2 70-80. 3. V. E. HoggattsJr. Fibonacci and Lucas Numbers* Boston: Houghton Mifflin9 1969. Pp. 34-35. 4. K. Stolarsky. "A Set of Generalized Fibonacc...

The generalized Fibonacci cube Qd(f) is the subgraph of the d-cube Qd induced on the set of all strings of length d that do not contain f as a substring. It is proved that if Qd(f) ∼= Qd(f ′) then |f | = |f ′|. The key tool to prove this result is a result of Guibas and Odlyzko about the autocorrelation polynomial associated to a binary string. An example of a family of such strings f , f ′, wh...