Binary Signed-Digit Integers, the Stern Diatomic Sequence and Stern Polynomials

追随 Richard Stanley 的脚步：与广义 Stern 二原子阵列相关的生成函数的自动生成

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。

Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

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…

通过广义 Chebyshev 多项式得到的 Stern 二原子序列

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。

Hyper $b$-ary expansions and Stern polynomials

We study a recently introduced base $b$ polynomial analog of Stern's diatomic sequence, which generalizes Stern polynomials of Klavar, Dilcher, Ericksen, Mansour, Stolarsky, and others. We lift some basic properties of base $2$ Stern polynomials to arbitrary base, and introduce a matrix characterization of Stern polynomials. By specializing, we recover some new number theoretic results about hyper $b$-ary partitions, which count partitions of $n$ into powers of $b$.

Record-Setters in the Stern Sequence

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.

A digit reversal property for an analogue of Stern's sequence

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 $n$.

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.

Largest values of the Stern sequence, alternating binary expansions and continuants

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…

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 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.

A digit reversal property for Stern polynomials

We consider the following polynomial generalization of Stern's diatomic series: let $s_1(x,y)=1$, and for $n\geq 1$ set $s_{2n}(x,y)=s_n(x,y)$ and $s_{2n+1}(x,y)=x\,s_n(x,y)+y\,s_{n+1}(x,y)$. The coefficient $[x^iy^j]s_n(x,y)$ is the number of hyperbinary expansions of $n-1$ with exactly $i$ occurrences of the digit $\mathtt 2$ and $j$ occurrences of $\mathtt 0$. We prove that the polynomials $s_n$ are invariant under \emph{digit reversal}, that is, $s_n=s_{n^R}$, where $n^R$ is obtained from $n$ by reversing the binary expansion of $n$.

Binary Signed-Digit Integers and the Stern Polynomial

The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in some depth. In this paper, we show previously unknown connections between BSD representations and the Stern polynomial. We derive a weight-distribution theorem for $i$-bit BSD representations of an integer $n$ in terms of the coefficients and degrees of the terms of the Stern polynomial of $2^i-n$. We then show new recursions on Stern polynomials, and from the…

非标准二进制表示与 Stern 序列

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。

Strong arithmetic property of certain Stern polynomials

Let $B_{n}(t)$ be the $n$th Stern polynomial, i.e., the $n$th term of the sequence defined recursively as $B_{0}(t)=0, B_{1}(t)=1$ and $B_{2n}(t)=tB_{n}(t), B_{2n+1}(t)=B_{n}(t)+B_{n-1}(t)$ for $n\in\N$. It is well know that $i$th coefficient in the polynomial $B_{n}(t)$ counts the number of hyperbinary representations of $n-1$ containing exactly $i$ digits 1. In this note we investigate the existence of odd solutions of the congruence \begin{equation*} B_{n}(t)\equiv 1+rt\frac{t^{e(n)}-1}{t-1}\pmod{m}, \end{equation*} where $m\in\N_{\geq 2}$ and $r\in\{0,\ldots,m-1\}$ are fixed and $e(n)=\op{…