The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. A generalization of these sequences is defined by Cm,s = ms m+1 and Wm,s = ms m−1, for s > 1. In this pa...

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a conne...

Among numerical sequences, the Fibonacci numbers have achieved a kind of celebrity status with such fabulous properties, it is no wonder that the Fibonacci numbers stand out as a kind of super sequence. Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is consequently, incredibly vast. The Fibonacci sequence has been generalized in a num...

We give an explicit formula for the figure of merit ρN of 2-dimensional rank 2 lattice rules in terms of continued fractions for rational numbers. Further we generalize Fibonacci lattice rules to rank 2 Fibonacci lattice rules which have the same ratio of the figure of merit to the number of points as the classical Fibonacci lattice rule.

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a conne...

To parallelize applications that require the use of random numbers, an efficient and good quality parallel random number generator is required. In this paper, we study the parallelization of lagged Fibonacci generators for distributed memory parallel computers. Two popular ways of generating a random sequence in parallel are studied: the contiguous subsequence technique and the leapfrog techniq...

Among numerical sequences, the Fibonacci numbers Fn have achieved a kind of celebrity status. Indeed, Koshy gushingly refers to them as one of the “two shining stars in the vast array of integer sequences” [16, p. xi]. The second of Koshy’s “shining stars” is the Lucas numbers, a close relative of the Fibonacci numbers, about which we will say more below. The Fibonacci numbers are famous for po...

For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n > 2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 = 1.61803398 . . . . But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by t1 = t2 = 1, and for n > 2, tn = ±tn−1 ± tn−2, where each ± sign is independent and eithe...

For a binary word f , let Qd(f) be the subgraph of the d-dimensional cube Qd induced on the set of all words that do not contain f as a factor. Let Gn be the set of words f of length n that are good in the sense that Qd(f) is isometric in Qd for all d. It is proved that limn→∞ |Gn|/2 exists. Estimates show that the limit is close to 0.08, that is, about eight percent of all words are good.

One says g is a Fibonacci primitive root modulo /?, wherep is a prime, iff g is a primitive root modulo/7 and g = g + 1 (mod p). In [1 ] , [2 ] , and [3] some interesting properties of Fibonacci primitive roots were developed. In this paper, we shall show that a necessary and sufficient condition for a prime/? ^ 5 to have a Fibonacci primitive root is p = 1 or 9 (mod 10) and Alp) = p 1, where/I...