Diatomic Sequence Research Articles

Asymptotic Analysis of q-Recursive Sequences

For an integer qge 2, a q-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q. In this article, q-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every q-recursive sequence is q-regular in the sense of Allouche and Shallit and that a q-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for q-recursive sequences are then obtained based on a general result on the asymptotic analysis of q-regular sequences.Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions ofStern’s diatomic sequence,the number of non-zero elements in some generalized Pascal’s triangle andthe number of unbordered factors in the Thue–Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.

Binary signed-digit integers and the Stern diatomic sequence

Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit representation of integers is an alternative representation of integers with much use in efficient computation, coding theory and cryptography. We link these two ideas here, showing that the number of $i$-bit binary signed-digit representations of an integer $n$ with $n<2^i$ is the $(2^i-n)^\text{th}$ element in Stern's diatomic sequence. This correspondence makes the vast range of results known for Stern's diatomic sequence available for consideration in the study of binary signed-digit integers.

Properties of Multivariate $${\varvec{b}}$$-Ary Stern Polynomials

Given an integer base $$b\ge 2$$, we investigate a multivariate b-ary polynomial analogue of Stern’s diatomic sequence which arose in the study of hyper b-ary representations of integers. We derive various properties of these polynomials, including a generating function and identities that lead to factorizations of the polynomials. We use some of these results to extend an identity of Courtright and Sellers on the b-ary Stern numbers $$s_b(n)$$. We also extend a result of Defant and a result of Coons and Spiegelhofer on the maximal values of $$s_b(n)$$ within certain intervals.

Palindromic Sequences of the Markov Spectrum

We study the periods of Markov sequences, which are derived from the continued fraction expression of elements in the Markov spectrum. This spectrum is the set of minimal values of indefinite binary quadratic forms that are specially normalised. We show that the periods of these sequences are palindromic after a number of circular shifts, the number of shifts being given by Stern's diatomic sequence.

Statistical distribution of the Stern sequence

We prove that the Stern diatomic sequence is asymptotically distributed according to a normal law, on a logarithmic scale. This is obtained by studying complex moments, and the analytic properties of a transfer operator.

A natural probability measure derived from Stern’s diatomic sequence

Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely singular continuous, with a strictly increasing, Holder continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence.

The Stern Diatomic Sequence via Generalized Chebyshev Polynomials

Let a(n) be the Stern diatomic sequence, and let x1,…, xr be the distances between successive 1's in the binary expansion of the (odd) positive integer n. We show that a(n) is obtained by evaluating generalized Chebyshev polynomials when the variables are given the values x1 + 1,…, xr + 1. We also derive a formula expressing the same polynomials in terms of sets of increasing integers of alternating parity and derive a determinant representation for a(n).

Upper Bounds for Stern's Diatomic Sequence and Related Sequences

Let $(s_2(n))_{n=0}^\infty$ denote Stern's diatomic sequence. For $n\geq 2$, we may view $s_2(n)$ as the number of partitions of $n-1$ into powers of $2$ with each part occurring at most twice. More generally, for integers $b,n\geq 2$, let $s_b(n)$ denote the number of partitions of $n-1$ into powers of $b$ with each part occurring at most $b$ times. Using this combinatorial interpretation of the sequences $s_b(n)$, we use the transfer-matrix method to develop a means of calculating $s_b(n)$ for certain values of $n$. This then allows us to derive upper bounds for $s_b(n)$ for certain values of $n$. In the special case $b=2$, our bounds improve upon the current upper bounds for the Stern sequence. In addition, we are able to prove that $\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}$.

Algebraic independence of the generating functions of the Stern polynomials and their twisted analogues

Dilcher and Stolarsky introduced and studied a polynomial analogue of Stern’s diatomic sequence. Similarly, we define here a polynomial analogue of Bacher’s twisted version of the Stern sequence. Our main aim is to consider the two-dimensional generating functions of both these polynomial sequences. More precisely, since they satisfy certain Mahler-type functional equations, we succeed in characterizing the algebraic independence over \({\mathbb{Q}}\) of the values of these two functions at algebraic points \({(\xi,\alpha)}\) with \({0 < |\xi|,|\alpha| < 1}\).

Hyperbinary Expansions and Stern Polynomials

We introduce an infinite class of polynomial sequences $a_t(n;z)$ with integer parameter $t\geq 1$, which reduce to the well-known Stern (diatomic) sequence when $z=1$ and are $(0,1)$-polynomials when $t\geq 2$. Using these polynomial sequences, we derive two different characterizations of all hyperbinary expansions of an integer $n\geq 1$. Furthermore, we study the polynomials $a_t(n;z)$ as objects in their own right, obtaining a generating function and some consequences. We also prove results on the structure of these sequences, and determine expressions for the degrees of the polynomials.

Zeros and convergent subsequences of Stern polynomials

We investigate Dilcher and Stolarsky’s polynomial analogue of the Stern diatomic sequence. Basic information is obtained concerning the distribution of their zeros in the plane. Also, uncountably many subsequences are found, each of which converges to a unique analytic function on the open unit disk. We thus generalize a result of Dilcher and Stolarsky from their second paper on the subject.

TRANSCENDENCE AND ALGEBRAIC INDEPENDENCE OF SERIES RELATED TO STERN'S SEQUENCE

In this same journal, Coons published recently a paper [The transcendence of series related to Stern's diatomic sequence, Int. J. Number Theory6 (2010) 211–217] on the function theoretical transcendence of the generating function of the Stern sequence, and the transcendence over ℚ of the function values at all non-zero algebraic points of the unit disk. The main aim of our paper is to prove the algebraic independence over ℚ of the values of this function and all its derivatives at the same points. The basic analytic ingredient of the proof is the hypertranscendence of the function to be shown before. Another main result concerns the generating function of the Stern polynomials. Whereas the function theoretical transcendence of this function of two variables was already shown by Coons, we prove that, for every pair of non-zero algebraic points in the unit disk, the function value either vanishes or is transcendental.

Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,…

Stern's diatomic sequence appeared in print in 1858 and has been the subject of numerous papers since. Our goal is to present many of these properties, both old and new. We present a large set of references and, for many properties, we supply simple proofs or ones that complement existing proofs. Among the topics covered are what these numbers count (hyperbinary representations) and the sequence's surprising parallels with the Fibonacci numbers. Quotients of consecutive terms lead to an enumeration of the rationals. Other quotients lead to a map from dyadic rationals to the rationals whose completion is the inverse of Minkowski's? function. Along the way, we get a distant view of fractals and the Riemann hypothesis as well as a foray into random walks on graphs in the hyperbolic plane.

Non-converging continued fractions related to the Stern diatomic sequence

Non-converging continued fractions related to the Stern diatomic sequence

A POLYNOMIAL ANALOGUE TO THE STERN SEQUENCE

We extend the Stern sequence, sometimes also called Stern's diatomic sequence, to polynomials with coefficients 0 and 1 and derive various properties, including a generating function. A simple iteration for quotients of consecutive terms of the Stern sequence, recently obtained by Moshe Newman, is extended to this polynomial sequence. Finally we establish connections with Stirling numbers and Chebyshev polynomials, extending some results of Carlitz. In the process we also obtain some new results and new proofs for the classical Stern sequence.

Hanoi graphs and some classical numbers

The Hanoi graphs H p n model the p-pegs n-discs Tower of Hanoi problem(s). It was previously known that Stirling numbers of the second kind and Stern's diatomic sequence appear naturally in the graphs H p n . In this note, second-order Eulerian numbers and Lah numbers are added to this list. Considering a variant of the p-pegs n-discs problem, Catalan numbers are also encountered.

Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence

It is known that in the Tower of Hanoi graphs there are at most two different shortest paths between any fixed pair of vertices. A formula is given that counts, for a given vertex v, the number of vertices u such that there are two shortest u, v-paths. The formula is expressed in terms of Stern’s diatomic sequence b( n) ( n≥0) and implies that only for vertices of degree two this number is zero. Plane embeddings of the Tower of Hanoi graphs are also presented that provide an explicit description of b( n) as the number of elements of the sets of vertices of the Tower of Hanoi graphs intersected by certain lines in the plane.

Carbonyl Boron and Related Systems: An ab Initio Study of B−X and YB⋮BY (1Σg+), Where X = He, Ne, Ar, Kr, CO, CS, N2 and Y = Ar, Kr, CO, CS, N2

Using the coupled-cluster methodology and large correlation consistent basis sets, we have examined the BX and YBBY (1Σg+) molecules, where X = He, Ne, Ar, Kr, CO, CS, and N2 and Y = Ar, Kr, CO, CS, and N2. For the B−X series we have constructed full potential energy curves reporting total energies, equilibrium geometries, binding energies, and also spectroscopic constants for the diatomic sequence. The B−CO, B−CS, and B−N2 ground states are of 4Σ- and 2Π symmetries, respectively, with the 4Σ- states having remarkably strong binding energies with respect to their adiabatic fragments. For all the triatomics the first excited 4P state of B plays an instrumental role in the binding process, while the bonding mechanism in either the 2Π or 4Σ- symmetries is due to charge transfer to the empty 2pz orbital of the B atom. The YBBY series results by singlet coupling two B−Y 4Σ- moieties, leading to acetylene-like YB⋮BY systems of 1Σg+ symmetry.