Linear combinations of factorials and $$S$$ S -units in a binary recurrence sequence

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.

A balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

On binary recurrence sequences

A Characterization of Certain Binary Recurrence Sequences

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial fin mathbb {Q}[x] can have at most one shift f(x)-lambda (lambda in mathbb {C}) of the form u(g(x))^q(h(x))^k with uin mathbb {C}, g,hin mathbb {C}[x] and either deg (g)=2, k is even, q=k/2 or deg (g)le 1, kge 2, qge 1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

Compact or efficient representation for either images or image sequences is key operation to performing image and video processing tasks, such as compression, analysis, etc. The efficiency of an approximation is evaluated by the sparsity measure of the approximation, i.e., the sparsest the representation is, the more efficient it is. For image processing tasks, it is often desired to decompose the image into a linear combination of few visual primitives or features selected from a large collection of waveforms, called the dictionary. These primitives are usually designed in such a way to achieve some good approximation performances by assuming a given class of functions to model images. A common class of functions for image modeling is the set of functions having discontinuities along smooth contours or boundaries, delineating smooth geometrical regions. It was shown that the well-known separable isotropically scaled two-dimensional wavelets fail to capture the geometrical regularity inherent to images. Motivated by this issue, we designed two rich dictionaries Ds and Dst, composed of multi-scaled ridge-like functions satisfying the anisotropy and the directionality features, for image and video expansions, respectively. Regarding the spatio-temporal dictionary Dst, we defined a warping operator W, which aligns these functions along the coherent motion trajectories in order to exploit the nature of the temporal evolution in the video signal. To obtain a compact representation for the visual signal over the dictionaries Ds and Dst, only the primitives that best match the signal components must be selected. This problem is well studied in approximation theory and it is known as the problem of sparse approximation in unrestricted dictionaries, whose optimal solution is NP-hard. Greedy approaches, such as MP and OMP, have been proposed to provide sub-optimal solutions by iteratively choosing one atom at a time. However, the computational complexity of these algorithms is cumbersome and limits their applicability. To alleviate this issue, we introduced a greedy algorithm, called the M-Term Pursuit MTP, whose performances are very close to those related to MP. Then we employed both algorithms, MP and MTP, for image and image sequence decompositions and we investigated their performances. A target application has been studied, which is the scalable compression, where the aim is to generate a bit-stream that can provide both SNR and geometrical scalability. Geometrical scalability refers to the spatial scalability in case the of images and to the spatio-temporal scalability in the case of video sequences. The SNR or quality scalability is related mainly to the nature of the greedy approaches and the embedded quantization and coding. To do so, we designed a rate allocation algorithm that offers a progressively refinable bit-stream, based on the subsets approach. On the other hand, the geometrical scalability is fullfiled thanks to the structure of the dictionaries, Ds and Dst, which are designed using parametric fashion. Comparisons with state-of-the-art scalable and non-scalable codecs illustrate the good performance of the proposed compression techniques at low rates.

This paper discusses the pulmonary washout tests and some of the considerations involved in applying optimal experimental design theory to these tests. Optimal inputs are described as binary sequences. For a given set of parameters the optimal input is determined by minimizing the determinant of the covariance matrix (J). Optimal inputs for a given set of parameters are determined using a modified random search procedure. The optimal input to deliver in the case where the parameters are unknown is determined by mimimizing a linear combination of the J's of several parameter sets which to span the range of interest. Individual J's are weighted by apriori probabilities and by associated costs. For the conditions studied in this work, the optimal inputs involved delivering tracer on breath 7 of a 10 breath test, 12,14,15 and 20 for a 20 breath test, 20,22,23,24 and 30 on a 30 breath test and breaths 23,25,26,32,34,35,40 of a 40 breath test. Implementation of these optimal inputs is also discussed.

Information systems are developing and improving very fast. The requirements for synchronization for such systems also increase. In such systems linear recurrent sequences are widely used for synchronization. In this paper filtration algorithms of linear recurrent sequences sequences are synthesized. The synthesized algorithms can be classified as quasi-noncoherent (subincoherent) algorithms, as opposed to quasi-coherent algorithms, which use phase estimate of the received signal, or to noncoherent algorithms that use averaging over the entire region of phase uncertainty. The results of computer simulation of developed algorithms for various signal-to-noise ratios and accumulation time are presented.

In the fields of communication and control there sometimes arises the problem of determining the characteristics of a time-invariant system from discrete records of its input and output during a limited interval of time, where the output data are contaminated with random noise. When this system is linear, we can use the convolution sum to obtain a characterization of the output as a linear combination of past inputs. Hill and McMurtry [4] showed that if we choose a Legendre binary noise sequence as input to a linear system, then the least squares approximation to the characterizing coefficients is expressible in a computationally feasible form, even when a large number of coefficients is involved. In this correspondence we characterize a nonlinear system by a generalization of the convolution sum and show that if we choose Golomb's maximal linear recurring binary-noise sequence [2] as input, then the least squares approximation to the characterizing coefficients is expressible in a computationally feasible form. Thus, the maximal linear recurring sequence occupies the same role in investigating certain nonlinear systems that the Legendre sequence occupies in investigating linear systems.

The chapter explores fundamental concepts of combinatorics, focusing on ordinary generating functions (FGOs) and their applications in counting problems. FGOs are powerful tools for analyzing sequences by transforming them into algebraic expressions. They help in solving problems related to permutations, combinations, and arrangements. The chapter demonstrates how FGOs can generate sequences like Fibonacci numbers and solve real-world problems like binary sequence formation. The chapter then discusses different types of permutations, including simple permutations, circular permutations, and permutations with repetition. It also covers arrangements, which are ordered selections of elements and their variations with repetition. Combinations, where order does not matter, are analyzed, particularly in games like poker and probability calculations. Special cases include combinations with repetition and dismutations, a concept where no element remains in its original position.

In this paper, we give explicit formulas for elements of the Fibonacci, and Lucas Pascal triangles. The structure of these objects and Pascal's original triangle coincide. Keeping the rule of addition, we replace both legs of the Pascal triangle by the Fibonacci sequence, and the Lucas sequence, respectively. At the end of the study we describe how to determine such a formula for any binary recurrence $\{G_n\}^{\infty}_{n=0}$ satisfying $G_n=G_{n−1}+G_{n−2}$. Other scattered results are also presented.

The k –generalized Fibonacci sequence $$(F_m^{(k)})_{m\ge 2-k}$$ ( F m ( k ) ) m ≥ 2 - k is the linear recurrent sequence of order k whose first k terms are $$0, \ldots , 0, 1$$ 0 , … , 0 , 1 and each term afterwards is the sum of the preceding k terms. The case $$k=2$$ k = 2 corresponds to the well known Fibonacci sequence. In Gómez and Luca (Lith. Math. J. 56(4):503–517, 2016), the multiplicative independence between terms of the same k -generalized Fibonacci sequence was studied. In this paper, we find all the multiplicative dependent pairs $$(F_m^{(k)},u_n)$$ ( F m ( k ) , u n ) where $$u_n$$ u n is a Fibonacci, a Lucas or a Pell number.

The best known linear recurrence sequence is the Fibonacci Sequence 1,1,2,3,5,8,..., where each term (after the first two terms) is a sum of the two preceding terms. We may extend it to become a “doubly infinite” sequence $$ ..., - 8,5, - 3,2, - 1,1,0,1,1,2,3,5,8,... . $$ .

In this automata column, Jöel Ouaknine and James Worrell beautifully write about beautiful recent work on linear recurrence sequences, which are sequences of integers defined by recursion like in the Fibonacci sequence. Clearly such sequences can appear almost everywhere, so there is no particularly good reason to talk about them in the automata column, but the authors politely point out such sequences can be used to count the number of words of given length accepted by an automaton. Thanks to this connection we get to have an automata column that cites Terrence Tao and uses logarithms of algebraic numbers.

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.