Fibonacci Type Sequences Research Articles

On the decomposition of the extensions of the Gini index that are based on the ‘metallic’ sequences of number theory

AbstractThe present paper extends the work of Subramanian (Econ Bull 41(4):2309–2319, 2021), as well as that of Creedy and Subramanian (Exploring a new class of inequality measures and associated value judgements: Gini and Fibonacci-type sequences, 2022), by showing how this new extension of the Gini index may be decomposed by income sources, income classes and population subgroups. It also gives an empirical illustration applying this extension of the Gini index to the analysis of the inequality in expenditure on social protection among European countries in 2018. In the decomposition by benefit function it appears that the housing function consistently exhibits the smallest contribution, due to its low expenditure weight, while the old-age function has the lowest inequality but the highest contribution, due to its substantial expenditure weight. The family/children function's contribution increases when a higher weight is given to lower expenditures, highlighting a pronounced inequality at lower expenditure levels. Conversely, the old-age function shows higher inequality among countries with higher expenditures. In the breakdown by welfare systems, we observe greater inequality between systems than within them.

New Fibonacci-type pulsated sequences

A new Fibonacci-type sequence from pulsated type is introduced. The explicit form of its members is given.

On Doubled and Quadrupled Fibonacci Type Sequences

Abstract In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana’s sequence. We give a binomial direct formula for these sequences using graph methods, and also we derive a number of identities. Moreover, we study matrix generators of these sequences and determine connections with the Pascal’s triangle.

Exploring A New Class of Inequality Measures and Associated Value Judgements: Gini and Fibonacci-Type Sequences

Exploring A New Class of Inequality Measures and Associated Value Judgements: Gini and Fibonacci-Type Sequences

Fibonacci type sequences and integer multiples of periodic continued fractions

We construct a class of quadratic irrationals having continued fractions of period \(n\geq2\) with `small' partial quotients for which specific integer multiples have periodic continued fractions with the length of the period being \(1\), \(2\) or \(4\), and with 'large' partial quotients. We then show that numbers in the period of the new continued fraction are functions of the numbers in the periods of the original continued fraction. We also show how polynomials arising from generalizations of these continued fractions are related to Chebyshev and Fibonacci polynomials and, in some cases, have hyperbolic root distribution.

A Fibonacci type sequence with Prouhet–Thue–Morse coefficients

Let , where is the sum of binary digits function. The sequence is the well-known Prouhet–Thue–Morse sequence. In this note we study the sequence , where and for we define recursively as follows: . We prove several results concerning arithmetic properties of the sequence . In particular, we prove non-vanishing of for , automaticity of the sequence for each m, and other results.

A short remark on a new Fibonacci-type sequence

A new Fibonacci-type sequence is constructed and for it is proved that it has a basis with 24 elements.

On Some Multinomial Sums Related to the Fibonacci Type Numbers

Abstract In this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.

A Two-Dimensional Gauss–Kuzmin Theorem Associated with the Random Fibonacci-Type Sequences

In this article, we attempt to investigate a two-dimensional Gauss–Kuzmin theorem for continued fraction expansions associated with random Fibonacci-type sequences introduced by Chan (2006). More precisely speaking, our focus is to obtain a Gauss–Kuzmin theorem concerning the natural extension of corresponding interval maps $$\{\tau _l: l\in \mathbb {N},l\ge 2\}$$ . Then, we give a local betterment of this theorem. It matters that together with characteristic properties of the Perron–Frobenius operator of $$\tau _l$$ under its invariant measure on the Banach space of functions of bounded variation, we are in a position to conclude unambiguous lower and upper bounds for the error term linked to distribution function in the case when $$2\le l\le 257$$ , which show that the desired optimal convergence rate is $$\mathcal {O}(\vartheta _l^n)$$ as $$n\rightarrow \infty $$ with $$\vartheta _l=\frac{2(l-1)}{2l-1+\sqrt{1+4(l-1)}}$$ .

On the golden number and Fibonacci type sequences

The paper presents, among others, the golden number $\varphi$ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein's theorem. To demonstrate the equality , where $f_n$ is $n$-th Fibonacci number also the formula from Corollary \ref{cor1} has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.

The n-dimensional Stern–Brocot tree

This paper generalizes the Stern–Brocot tree to a tree that consists of all sequences of [Formula: see text] coprime positive integers. As for [Formula: see text] each sequence [Formula: see text] is the sum of a specific set of other coprime sequences, its Stern–Brocot set [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in [Formula: see text] and a multiresolution partition of [Formula: see text]. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern–Brocot set forms the matrix columns. This induces a finitely generated submonoid [Formula: see text] of [Formula: see text], and a unimodular multidimensional continued fraction algorithm, also generalizing [Formula: see text]. It turns out that the [Formula: see text]-dimensional subtree starting with a sequence [Formula: see text] is isomorphic to the entire [Formula: see text]-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation.

Relationships between Fibonacci-type sequences and Golden-type ratios

Relationships between Fibonacci-type sequences and Golden-type ratios

Fast elliptic curve algorithm using deformed Fibonacci-type series

The efficient and secure scalar multiplication can be constructed by combining the addition chain and elliptic curve. In this paper, we propose an explicit algorithm which is a kind of 'double-addition' addition chain based on the Fibonacci-type sequence. The 'double-addition' sequence of arbitrary integer k is calculated by using Fibonacci and golden ratio addition chain method. In addition, each cycle of this method is fixed to perform double point and a point add operations based on the 'double-addition' of the elliptic curve scalar multiplication algorithm, so as to be able to resist the simple power attack. At the same time, the length of the chain is greatly shortened by using the double point operation compared with the chain of the Fibonacci. Experimental results showed that the average chain length has attained 38% to 55% reduction compared to other doubling-free addition chain methods.

Fast elliptic curve algorithm using deformed Fibonacci-type series

The efficient and secure scalar multiplication can be constructed by combining the addition chain and elliptic curve. In this paper, we propose an explicit algorithm which is a kind of 'double-addition' addition chain based on the Fibonacci-type sequence. The 'double-addition' sequence of arbitrary integer k is calculated by using Fibonacci and golden ratio addition chain method. In addition, each cycle of this method is fixed to perform double point and a point add operations based on the 'double-addition' of the elliptic curve scalar multiplication algorithm, so as to be able to resist the simple power attack. At the same time, the length of the chain is greatly shortened by using the double point operation compared with the chain of the Fibonacci. Experimental results showed that the average chain length has attained 38% to 55% reduction compared to other doubling-free addition chain methods.

On a new improved unifying closed formula for all Fibonacci-type sequences and some applications

In this paper, we establish a new general framework under which all generalized Fibonacci-type sequences come together. The main result lies in obtaining one new general closed formula solution for all such sequences by using only matrix theory. This new formula in turn gives new closed formula solutions for most well-known sequences of this type. Next, the advantages of this formula are exploited. In particular, two applications are presented; one deals with finding the most general congruence formula for such sequences which generalizes most known formulas of this type and the other is concerned with computational number theory.

Non-standard Aspects of Fibonacci Type Sequences

Fibonacci sequence and the limit of the quotient of adjacent Fibonacci numbers, namely the Golden Mean, belong to general knowledge of almost anybody, not only of mathematicians and geometers. There were several attempts to generalize these fundamental concepts which also found applications in art and architecture, as e.g. number series and quadratic equations leading to the so-called ``Metallic means'' by V. de Spinadel [8] or the cubic ``plastic number'' by van der Laan [5] resp. the ``cubi ratio'' by L. Rosenbusch [7]. The mentioned generalisations consider series of integers or real numbers. ``Non-standard aspects'' now mean generalisations with respect to a given number field or ring as well as visualisations of the resulting geometric objects. Another aspect concerns Fibonacci type resp. Padovan type combinations of given start objects. Here it turns out that the concept ``Golden Mean'' or ``van der Laan Mean'' also makes sense for vectors, matrices and mappings.

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.

A General Family of Fibonacci-Type Squences

In this work, we introduce a further generalization of the Fibonacci-type sequence, namely, generalized Fibonacci-type sequence. We also provide the general solution of nonhomogeneous generalized Fibonacci-type sequence, which can be expressed in terms of the Fibonacci-type numbers.

NEW PERSPECTIVES IN THE METRIC THEORY OF CONTINUED FRACTION EXPANSION RELATED TO FIBONACCI TYPE SEQUENCES

A survey of the metric theory of the continued fraction expansions related to random Fibonacci Type sequences discussed by Sebe and Lascu is given. The limit properties of these expansions have been studied. A Wirsing-type approach to the Perron-Frobenius operator of the generalized Gauss map under its invariant measure allows us to get close to the optimal convergence rate. Actually, we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-Levy problem for these expansions.

Aberrant phyllotactic patterns in cones of some conifers: a quantitative study

The scale patterns of 6000 cones from one single tree of &lt;em&gt;Pinus nigra&lt;/em&gt; Arn. have been examined. Apart from the main Fibonacci pattern with 8 and 13 parastichies, nine aberrant spiral patterns with Fibonacci-type sequences have been found. They are quite rare and occur with different frequencies. The parastichy quotient 8/13 of the prevalent pattern is very close to the golden ratio 0.618. In case of the black pine it appeared that the greater the deviation of the parastichy quotient &lt;em&gt;m&lt;/em&gt;/&lt;em&gt;n&lt;/em&gt; from 0.618, the rarer the pattern. Similar results obtained for the sample of 1506 cones collected from three individual trees of larch (&lt;em&gt;Larix decidua&lt;/em&gt; Mill.) suggest a true correlation between the frequency of a pattern and the deviation of its parastichy quotient from the golden ratio.