Motivated by recent work of Trümper, we consider the general Collatz word (updown pattern) and the sequences following this pattern. We derive recurrences for the first and last sequence entries from repeated application of the general solution of a binary linear inhomogeneous Diophantine equation. We solve these recurrences and also discuss the Collatz tree.

We derive recurrences for counting the number a(n, r) of sequences of length n with Lempel-Ziv complexity r, which has important applications, for instance testing randomness of binary sequences. We also give algorithms to compute these recurrences. We employed these algorithms to compute a(n, r) and expected value, EPn, of number of patterns of a sequence of length n, for relatively large n. W...

In this paper we derive factorizations and representations of a polynomial analogue of an arbitrary binary sequence by matrix methods. It generalizes various results on Fibonacci, Lucas, Chebyshev and Morgan–Voyce polynomials. 1. Introduction In [10], the divisibility properties of the Fibonacci polynomial sequence ffn (x)g was studied. The Fibonacci polynomial sequence is de…ned by the recursi...

Let Rn (n = 0, 1, 2, . . .) be a second order linear recursive sequence of rational integers defined by Rn = ARn−1+BRn−2 for n > 1, where A and B are integers and the initial terms are R0 = 0, R1 = 1. It is known, that if α, β are the roots of the equation x2 −Ax−B = 0 and |α| > |β|, then Rn+1/Rn −→ α as n −→ ∞. Approximating α with the rational number Rn+1/Rn, it was shown that ∣∣∣α− Rn+1 Rn ∣...

We use a refined version of Roth’s Lemma, proved with the help of Faltings’ Product Theorem, in order to give an upper bound for the multiplicity of a binary linear recurrence. Résumé Nous utilisons une version du Lemme de Roth, provenant du Théorème du Produit de Faltings, pour majorer la multiplicité d’une récurrence linéaire binaire. Version française abrégée Dans l’article [7] H.-P. Schlick...

In this paper, we look at a diophantine equation of the form un = ( x y ) , where (un)n≥0 is a binary recurrent sequence of integers. We show that if the pair of integers (x, y) belongs to a fixed line of the Pascal triangle, then the above equation has only finitely many positive integer solutions (n, x, y). A binary recurrent sequence of integers (un)n≥0 has u0, u1 ∈ Z and satisfies the recur...

Abstract There are many results in the literature concerning linear combinations of factorials among terms recurrence sequences. Recently, Grossman and Luca provided effective bounds for such binary In this paper we show that under certain conditions, even greatest prime divisor $$u_n-a_1m_1!-\dots -a_km_k!$$ <mml:m...

Background: Conflicting studies link several conditions and risk factors to Dupuytren’s disease (DD). A questionnaire-based case-control study was set to investigate associated conditions and clinical features of DD in a sample of Italian patients. The main purpose was the identification of predicting factors for: DD development; involvement of multiple rays; involvement of both hands; developm...