Related papers: Integers represented by binary recursive sequences
A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the…
We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a given bound.
In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…
Given a negative $D>-(\log X)^{\log 2-\delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…
The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to…
In this paper we consider the problem of determining the law of binary stochastic processes from transition kernels depending on the whole past. These kernels are linear in the past values of the process. They are allowed to assume values…
Sequence representations supporting not only direct access to their symbols, but also rank/select operations, are a fundamental building block in many compressed data structures. Several recent applications need to represent highly…
In this paper we consider the sequence whose n^{th} term is the number of h-vectors of length n. We show that the n^{th} term of this sequence is bounded above by the n^{th} Fibonacci number and bounded below by the number if integer…
It is proven that, contrarily to the common belief, the notion of zero is not necessary for having positional representations of numbers. Namely, for any positive integer $k$, a positional representation with the symbols for $1, 2, \ldots,…
A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for…
The critical exponent of an infinite word $\bf x$ is the supremum, over all finite nonempty factors $f$, of the exponent of $f$. In this note we show that for all integers $k\geq 2,$ there is a binary infinite $k$-automatic sequence with…
Sequences whose terms are equal to the number of functions with specified properties are considered. Properties are based on the notion of derangements in a more general sense. Several sequences which generalize the standard notion of…
In this paper a class of simple, but nonlinear, systems of recursions involving $2$ dependent variables $x_{j}\left( n\right) $ is identified, such that the solutions of their initial-values problems -- with arbitrary initial data…
If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We…
We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.
Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words.…
How we choose to represent our data has a fundamental impact on our ability to subsequently extract information from them. Machine learning promises to automatically determine efficient representations from large unstructured datasets, such…
We define impulse response sequence in the set of all linear recurring sequences satisfying a linear recurrence relation of order $r$. The generating function and expression of the impulse response sequence are presented. Some identities of…