Yasuhisa Yamada
We define a function by refining Stern's diatomic sequence. We name it the {\it assembly function}. It is strictly increasing continuous. The first and the second main theorems are on an action to the function. The third theorem is on differentiability of the function at rational points.
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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, Hölder continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence.
This paper presents the construction and various properties of substitution tilings related to Stern's diatomic series and based on tiles decorated with elements of $\mathbb{F}_p$ for some odd prime number $p$. These substitution tilings are additive in a sense that will be clarified later and lead to new nonperiodic tilings in one and two dimensions.
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 s…
Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit (BSD) representation of integers is used widely in efficient computation, coding theory and other applications. We link these two objects, showing that the number of $i$-bit binary signed-digit representations of an integer $n<2^i$ is the $(2^i-n)^\text{th}$ element in Stern's diatomic sequence. This correspondence makes the vast range of results known about the Stern diatomic sequence available for consideration in the study of binary signed-digit integers, an…
We answer a question of Calkin and Wilf concerning the maximal order of Stern's diatomic sequence. Specifically, we prove that $$\limsup_{n\to\infty}\frac{a(n)}{φ^{\log_2 n}}=\frac{φ^{\log_2 3}}{\sqrt{5}},$$ where $φ=(\sqrt{5}+1)/2$ is the golden ratio. This improves on previous results given by Berlekamp, Conway, and Guy, who showed that the limit value was bounded above by 1.25, and by Calkin and Wilf, who showed that the exact value was in the interval $[(φ/{\sqrt{5}})({3}/{2})^{\log_2φ},(φ+1)/{\sqrt{5}}].$
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular multidimensional continued fraction algorithm (the Farey algorithm), we will generalize the diatomic sequence to a collection of numbers that quite naturally should be called the triatomic sequence (or a two-dimensional Pascal with memory sequence). As continued fractions and the diatomic sequence can be thought of as coming from systematic subdivisions of the unit interv…
The Stern diatomic sequence via generalized Chebyshev polynomials
本文研究 Stern 二原子序列，证明其在变量取特定值时可通过求值广义 Chebyshev 多项式得到，并推导出用奇偶性交替递增整数集合表示多项式的公式，同时给出了基于单位矩阵与特定三对角矩阵之和的行列式表达式。
Automated Generation of Generating Functions Related to Generalized Stern's Diatomic Arrays in the footsteps of Richard Stanley
本文提出并实现了基于符号动态规划的 Maple 算法，用于自动生成广义 Stern 阵列的生成函数，并将其推广到由常系数线性递推序列定义的更一般阵列。
The Stern diatomic sequence is closely linked to continued fractions via the Gauss map on the unit interval, which in turn can be understood via systematic subdivisions of the unit interval. Higher dimensional analogues of continued fractions, called multidimensional continued fractions, can be produced through various subdivisions of a triangle. We define triangle partition-Stern sequences (TRIP-Stern sequences for short), higher-dimensional generalizations of the Stern diatomic sequence, from the method of subdividing a triangle via various triangle partition algorithms. We then explore seve…
We consider a variant of Stern's diatomic sequence, studied recently by Northshield. We prove that this sequence $b$ is invariant under \emph{digit reversal} in base $3$, that is, $b_n=b_{n^R}$, where $n^R$ is obtained by reversing the base-$3$ expansion of $n$.
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.
We study the largest values of the $r$th row of Stern's diatomic array. In particular, we prove some conjectures of Lansing. Our main tool is the connection between the Stern sequence, alternating binary expansions and continuants. This allows us to reduce the problem of ordering the elements of the Stern sequence to the problem of ordering continuants. We describe an operation that increases the value of a continuant, allowing us to reduce the problem of largest continuants to ordering continuants of very special shape. Finally, we order these special continuants using some identities and ine…
Stern's diatomic series, denoted by $(a(n))_{n \geq 0}$, is defined by the recurrence relations $a(2n) = a(n)$ and $a(2n + 1) = a(n) + a(n + 1)$ for $n \geq 1$, and initial values $a(0) = 0$ and $a(1) = 1$. A record-setter for a sequence $(s(n))_{n \geq 0}$ is an index $v$ such that $s(i) < s(v)$ holds for all $i < v$. In this paper, we give a complete description of the record-setters for the Stern sequence.
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 which each converge 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.
1 : We use properties of the Stern Sequence for numerical computations of moments $\int^1_0 t^n d?(t)$ associated to Minkowski's Question Mark function.
In a recent paper, Roland Bacher conjectured three identities concerning Stern's sequence and its twist. In this paper we prove Bacher's conjectures. Possibly of independent interest, we also give a way to compute the Stern value (or twisted Stern value) of a number based solely on its binary expansion.
We describe a twisted version of the Stern sequence and study a few of its properties.
In a recent preprint on ArXiv, Bacher introduced a twisted version of the Stern sequence. His paper contains in particular three conjectures relating the generating series for the Stern sequence and for the twisted Stern sequence. Soon afterwards Coons published two papers in {\it Integers}: first he proved these conjectures, second he used his result to obtain a correlation-type identity for the Stern sequence. We recall here a simple result of Reznick and we state a similar result for the twisted Stern sequence. We deduce an easy proof of Coons' identity, and a simple proof of Bacher's conje…
We present the classical Stern-Brocot tree and provide a new proof of the fact that every rational number between 0 and 1 appears in the tree. We then generalize theStern-Brocot tree to allow for arbitrary choice of starting terms, and prove that in all cases the tree maintains the property that every rational number between the two starting terms appears exactly once.
Farey's sequence is a well-known procedure used to generate proper fractions from 0 to 1. Farey sequence is commonly used in rational approximations of irrational numbers, ford circles and in Riemann hypothesis. Thus, in this paper, we aim to use properties of the Farey's sequence to prove the popular gcd theorem.
We prove a conjecture of Northshield by determining the maximal order of his analogue of Stern's sequence for $\mathbb{Z}[\sqrt{2}]$. In particular, if $b$ is Northshield's analogue, we prove that $$\limsup_{n\to\infty}\frac{2b(n)}{(2n)^{\log_3 (\sqrt{2}+1)}}=1.$$
Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。
We employ infinite ergodic theory to show that the even Stern-Brocot sequence and the Farey sequence are uniformly distributed mod 1 with respect to certain canonical weightings. As a corollary we derive the precise asymptotic for the Lebesgue measure of continued fraction sum-level sets as well as connections to asymptotic behaviours of geometrically and arithmetically restricted Poincaré series. Moreover, we give relations of our main results to elementary observations for the Stern-Brocot tree.
Let $B_{n}(t)$ be a $n$-th Stern polynomial and let $e(n)=\op{deg}B_{n}(t)$ be its degree. In this note we continue our study started in \cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence $\{e(n)\}_{n=1}^{\infty}$. We also study the sequence $d(n)=\op{ord}_{t=0}B_{n}(t)$. Among other things we prove that $d(n)=ν(n)$, where $ν(n)$ is the maximal power of 2 which dividies the number $n$. We also count the number of the solutions of the equations $e(m)=i$ and $e(m)-d(m)=i$ in the interval $[1,2^{n}]$. We also obtain an interesting closed expression for a …
Let w be a binary string and let a_w (n) be the number of occurrences of the word w in the binary expansion of n. As usual we let s(n) denote the Stern sequence; that is, s(0)=0, s(1)=1, and for n >= 1, s(2n)=s(n) and s(2n+1)=s(n)+s(n+1). In this note, we show that s(n) = a_1 (n) + \sum_{w in 1 (0+1)*} s([w bar]) a_{w1} (n) where w bar denotes the complement of w (obtained by sending 0 to 1 and 1 to 0, and [w] denotes the integer specified by the word w interpreted in base 2.
Non-standard binary representations and the Stern sequence
本文证明整数的短二进制有符号数字表示数量等于 Stern 序列对应项，并以直接、双射、生成函数及 Monroe 的结果给出多种论证。
The Stern polynomials defined by $s(0;x)=0$, $s(1;x)=1$, and for $n\geq 1$ by $s(2n;x)=s(n;x^2)$ and $s(2n+1;x)=x\,s(n;x^2)+s(n+1;x^2)$ have only 0 and 1 as coefficients. We construct an infinite lower-triangular matrix related to the coefficients of the $s(n;x)$ and show that its inverse has only 0, 1, and $-1$ as entries, which we find explicitly. In particular, the sign distribution of the entries is determined by the Prouhet-Thue-Morse sequence. We also obtain other properties of this matrix and a related Pascal-type matrix that involve the Catalan, Stirling, Fibonacci, Fine, and Padovan n…
Sums with Stern-Brocot sequences and Minkowski question mark function
本文肯定回答了 Moshchevitin 关于 Stern-Brocot 序列余项在 n 趋于无穷时是否趋于 0 的问题，并给出了关于 Minkowski 问号函数逆函数 Fourier-Stieltjes 系数定理的正确证明与扩展结果。
Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-λ_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $λ_{n+1}/λ_n > 2$. It is known that the denominators $Q_n(X)$ of the convergents of its continued fraction expansion are polynomials with coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is the $n$th term of the Stern-Brocot sequence. We show that replacing the index $n$ by any 2-adic integer $ω$ makes sense. We prove that $Q_ω(X)$ is a polynomial if and only if $ω\in {\mathbb Z}$. In all the other cases $Q_ω(X)$ is an infinite formal powe…
In this paper we study a natural generalization of the Stern-Brocot sequences which comes from the introduction of weighted mediants. We focus our attention on the case $k = 3$, in which $(2a + c)/(2b + d)$ and $(a + 2c)/(b + 2d)$ are the two mediants inserted between $a/b$ and $c/d$. We state and prove several properties about the cross-differences of Stern-Brocot sequences with $k = 3$, and give a proof of the fractal-like rule that describes the cross-differences of the unit $k = 3$ Stern-Brocot sequences, i.e. the one with usual starting terms $0/1, 1/1$ and with reduction of fractions.
We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: $B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)$, and $B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)$. We study also the sequence $e(n)=\op{deg}_{t}B_{n}(t)$ and give various of its properties.
The Stern sequence (s(n)) is defined by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1). Stern showed in 1858 that gcd(s(n),s(n+1)) = 1, and that for every pair of relatively prime positive integers (a,b), there exists a unique n so that s(n) = a and s(n+1) = b. We show that, in a strong sense, the average value of s(n)/s(n+1) is 3/2, and that for all d, (s(n),s(n+1)) is uniformly distributed among all feasible pairs of congruence classes modulo d. More precise results are presented for d = 2 and 3.
Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are useful for enumerating the rational numbers.
Zeros of Stern polynomials in the complex plane
本文研究Stern多项式序列在复平面上的零点分布，利用复连分数收敛性的Parabola定理证明圆盘$|w-2|\leq 1$内无根，从而部分推进Dilcher等人的猜想，并由此证明素数指标下Stern多项式在整数环上不可约。
Parity vectors and paradoxical sequences in the accelerated Collatz map
本文研究了加速科拉茨映射中奇偶向量和悖论序列的性质，通过三个定理和一个数值观察，提出了关于奇偶向量密度、悖论序列计数以及有界长度悖论序列密度零的结论。
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