Hyperbolic Fibonacci Sequence

We construct the sequences of Fibonacci and Lucas in any quadratic field $\mathbb{Q}(\sqrt{d}\,)$ with $d>0$ square free, noting that the general properties remain valid as those given by the classical sequences of Fibonacci and Lucas for the case $d = 5$, under the respective variants. For this construction, we use the fundamental unit of $\mathbb{Q}(\sqrt{d}\,)$ and then we observe the generalizations for any unit of $\mathbb{Q}(\sqrt{d}\,)$. Under certain conditions some of these constructions correspond to $k$-Fibonacci sequence for some $k\in \mathbb{N}$. Further, for both sequences, we obtain the generating function, Golden ratio, Binet's formula and some identities that they keep.

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

In studies presented in the literature, relationships between music and mathematics can sometimes be observed. Leonardo Fibonacci (1170-1250) is well known with the Fibonacci Sequence used to identify numbers in various music elements. In related studies, these numbers have been used to demonstrate the existence of the ‘Golden Ratio’ using methods and theories borrowed from the components of music. Nevertheless, this relationship has subsequently been seen inaccurate. The selected works from Chopin, Mozart, Beethoven and Bartók have previously been used to study the Fibonacci Sequence and Golden Ratio, and these works are discussed in this paper in the context of musical and mathematical theories. For musical works that allegedly used the Fibonacci Sequence and the Golden Ratio, we examined the measure or rhythms (sound duration)within these works; we found these studies yielded values close to the terms of the Fibonacci Sequence, and the determined values of the Golden Ratio were 0.618, 1.618, and 0.382. It was determined that the accuracy of the Fibonacci Sequence and the Golden Ratio expressed in selected works of European art music are controversial within the framework of relevant studies.

The aim of this investigation was to examine the relationship between the golden ratio and atomic structure. The chart of nuclides was evaluated for evidence of a true golden ratio. That is, the neutron number divided by the proton number needed to equal 1.618 and the atomic number (neutrons plus protons) divided by the neutron number needed to equal 1.618. Across the chart of nuclides, this ratio approximated the golden ratio to a varying degree of accuracy for some nuclides but rarely for both calculations. This relationship (golden ratio in the chart of nuclides) has not been previously reported. It was determined that specific nuclides exhibit the golden ratio between protons and neutrons, and that these produce a trend line on the chart of nuclides. The Fibonacci sequence is mathematically evident and this information can be used to postulate new nuclides or elements.

The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.

The aim of this investigation was to examine the relationship between the golden ratio and atomic structure. The chart of nuclides was evaluated for evidence of a true golden ratio. That is, the neutron number divided by the proton number needed to equal 1.618 and the atomic number (neutrons plus protons) divided by the neutron number needed to equal 1.618. Across the chart of nuclides, this ratio approximated the golden ratio to a varying degree of accuracy for some nuclides but rarely for both calculations. This relationship (golden ratio in the chart of nuclides) has not been previously reported. It was determined that specific nuclides exhibit the golden ratio between protons and neutrons, and that these produce a trend line on the chart of nuclides. The Fibonacci sequence is mathematically evident and this information can be used to postulate new nuclides or elements.

BackgroundThe Golden Ratio (GR) and the Fibonacci sequence have wide applications in biodiversity research, and recent studies indicate that the GR can be highlighted in the organization and physiological functioning of many body systems. The aim of this cross-sectional descriptive study is to determine the applicability of a mathematical model derived from the Fibonacci sequence to investigate the changes in hand grip strength (HGS) induced by the aging process.MethodsWe assessed the HGS for both hands, using a Saehan hydraulic hand dynamometer in a group of autonomous elderly subjects. One hundred twenty 55-year-old subjects (58 males and 62 females) and seventy 89-year-old subjects (31 men and 39 women) were included in the study group. All subjects were completely independent or independent with minimal assistance in activities of daily living (ADL), as determined after applying the Barthel index of ADL. The data series were statistically processed using descriptive statistics (univariate analysis) and inferential statistical methods (the t test for unpaired groups, with effect size measure – Cohen’s d and the ratio of the means method).ResultsThe decline of the relative HGS between the two age groups can be expressed by values close to the GR value (p < 0.001), both in relation to body symmetry (left hand/right hand evaluation) and laterality (dominant hand/non-dominant hand evaluation), for both sexes. For the whole group of men and women, the rhythm of HGS decline may be expressed by a value (1.61) notably close to the GR, regardless of body symmetry or laterality.ConclusionsThe common pattern of the relative HGS reduction between 55 and 89 years, as expressed by a value notably close to GR, can be considered to be an expression of a specific and predictable manifestation of the aging process, in the absence of disability.

The study mainly focuses on the use of the Golden Ratio and the Fibonacci sequence. The connection between them can be clearly visible in nature. With the help of the Fibonacci sequence scientists have solved many mysteries related to nature. Everything that is around us somehow or other depends on Fibonacci numbers, the Golden Ratio, and the Fibonacci sequence. Some examples are –Flower petals- Lily, Rose, Daisy, Marigold, Sunflower, Iris, Buttercups, wild rose, larkspur Trillium, Bloodroot, Aster, and Susan; Seed heads-Sunflower; Snail; Fruit-Apple, Banana, Pineapple; Human Face; Tree Branches; Cyclone; Pinecones; Shells; Spiral Galaxies; Bees; Famous architecture design – Taj Mahal, in Hindu rituals, in decoding-coding the data, in providing security to the sensitive data and all over the world, in mother's womb (about her baby's position), etc. The current study reflects that there is no limitation to the Fibonacci pattern and Golden Ratio in our surroundings.

In recent years, the scientific community has shown renewed interest in the Fibonacci numbers and in the Golden Section. The American-based Fibonacci Association is largely devoted to number theory [11,29] while the Slavonic group of the Fibonacci scientists is pursuing applications in philosophy, architecture, biology, computer science, and physics as reported in annual international seminars entitled “The Golden Section and Problems of the System Harmony” since 1992. This article discusses contemporary investigations of the Slavonic Fibonacci scientists with representative references [2–4,10,13–28,30,31].

The golden proportion is widely believed to be extraordinarily prevalent in nature and the arts, which is often ascribed to it being the limit of the ratio between any two successive elements in the Fibonacci sequence. It is suggested here that the golden ratio may not be as exceptional as generally believed. Mathematically, some interesting properties are common to all members of a family of sequences, denoted ARS, characterised as solutions to the classic rabbit reproduction problem varying on some parameter, j, including the Fibonacci sequence as a prototypical member--ARS2. Furthermore, for j > 1, any limit of the ratio between successive elements in ARS(j), shares the same formal properties with all other such limits. Three actual interpretations and three further geometric applications of ARS3, all intimately analogous to corresponding ARS2 ones, are presented for the sake of illustration. Empirically, it is suggested here that, owing to the communality of interesting mathematical properties between ARS sequences, as well as between corresponding limits, nature might appear to have made use of some other limits, aside of its variegated use of the limit of ARS2--the golden ratio. Initial empirical clues are provided. Finally, the issue whether there really is special import to golden proportions in nature and the arts is revisited in view of some empirical comparisons of appearances related to Fibonacci numbers and ARS3 numbers, particular its limit (-1.466) and the inverse of that limit (-0.682). It is argued that the claim that Fibonacci-related numbers are especially distinguished seems to warrant a more qualified approach than it has often met.

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform.

The article investigates the use of Fibonacci numbers and «golden sections» of Archimedes and Pythagoras in their use in architecture, art, painting, engineering, medicine, chemistry, physics, mathematics and other spheres of human life. It is also known that in the world around us all nature is built and develops on the principles of the golden ratio of Fibonacci numbers and the golden ratio of Archimedes and Pythagoras in their increasing and decreasing series. It is also known that the whole and its parts in the living world are also built on the principle of the golden ratio. Scientists have also proveт that our eye and its structure are also «designed» by nature on the basis of the golden ratio. Data transmission, which perceives the human eye is transmitted to the human brain waves also on the principles of the golden ratio. That is, what surrounds us and how we built in nature is built on the principles of the golden ratio. That is, our life is based on the principles of the golden ratio. Everything that man creates to make it rational, natural, beautiful and we like should also be built on the principles of the golden ratio and Fibonacci numbers. Since the information that we perceive by 90% passes through the eyes, and what we perceive most (about 70%) is the form, and the other 20% are colours, signs, drawings, inscriptions, in order to properly influence a person naturally and especially for «green» brands and TMs, which can be transmitted through packaging, the product or service itself, or communication, or presentation or in any other way, we must use Fibonacci numbers and golden sections. It is proposed to use the golden ratio in marketing when planning and managing an enterprise, in particular for calculating and forecasting prices, the appearance and size of packaging, as well as developing color schemes for packaging, trademarks, brands, placing goods on shelves, the shape and placement of labels on packaging and everything else related to the shape, size, color schemes, signs and symbols; calculating and looking at shelving inside and outside points of sale; developing a communications plan and implementing it, developing and managing musical accompaniment, smells and other means, which improves perception and communication, ranging from individual brands to product brands, companies, distributors, customer brands and «green brands».

This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.

ABSTRACTPhyllotaxis, the arrangement of leaves around a stem, shows in the vast majority of cases a regularity in the divergence angle of subsequent leaves which divide the whole circle into regular fractions. These are in most cases rational fractions of two Fibonacci numbers in an alternating series, converging towards the irrational limit of the golden section, corresponding to the golden divergence angle of 137.5 . . . degrees. This peculiarity was a long‐standing mystery. Here, it is related to the evolutionary pressure of optimal light capture for maximal photosynthetic activity. A model is established which relates minimal shadowing for the lower leaves to the divergence angle. Numerical results of this model agree well with semi‐empirical data on the dependence of light capture from the divergence angle. The basic shadow function of the model is also related with the demand of minimal shadowing for the angular separation of leaves and obtain, using elementary number theory, as solution the golden section. Further numerical studies show that the rational approach to the golden section (Schimper–Braun series) is related to the leaf width and the number of leaves of the plant.

The construction of the regular pentagon has always meant a difficult geometrical exercise for architects during the Middle Ages. As the correct drafting was forgotten after the Antiquity, several methods for its approximation had been invented in medieval times. As Golden Ratio appears between several parts of the regular pentagon, the role of the Fibonacci sequence in these approximate constructions is to be examined. The pentagonal drawing in the sketchbook of Villard de Honnecourt calls our attention to a possible way how medieval architects could have applied simple numerical ratios for getting angles they needed. The approximation of 72°, for instance, is likely to have been crucial for this pentagonal construction, as well as the approximation of Golden Ratio that could have been achieved by neighbouring pairs from Fibonacci’s sequence.