Related papers: How does the Gerrymander Sequence Continue?
In an earlier paper we introduced the notion of 'bifurcating continued fractions' in a heuristic manner. In this paper a formal theory is developed for the 'bifurcating continued fractions'.
We use the generating function of the characters of $C_2$ to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple Lie algebra. From this generating function we derive…
Several continued fraction expansions for $e$ have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven.…
We prove an existing conjecture that the sequence defined recursively by $a_1=1, a_2=2, a_n=4a_{n-1}-2a_{n-2}$ counts the number of length-$n$ permutations avoiding the four generalized permutation patterns 1-32-4, 1-42-3, 2-31-4, and…
Divide-and-conquer functions satisfy equations in F(z),F(z^2),F(z^4)... Their generated sequences are mainly used in computer science, and they were analyzed pragmatically, that is, now and then a sequence was picked out for scrutiny. By…
In this paper the approach to obtaining nonrecurrent formulas for some recursively defined sequences is illustrated. The most interesting result in the paper is the formula for the solution of quadratic map-like recurrence. Also, some…
We investigate and generalise Levine sequences like A011784, A061892 and A061894 and develop an algebraic theory for them. We thereby also cover other fast growing sequences like A014644, which we call golombic due to their strong ties with…
The replicator equation is interpreted as a continuous inference equation and a formal similarity between the discrete replicator equation and Bayesian inference is described. Further connections between inference and the replicator…
The basis of this work is a simple, extended corollary of Wilson's theorem. This corollary generates many more quotients than those already generated by Wilson's theorem, and it was of interest to derive how they relate to each other and…
This paper is a sequel to [3]. We formulate a natural algebraic geometry conjecture, give some of its number theoretic and analytical consequences, and show that those can be used to get further advances in wave turbulence theory.
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…
The purpose of this paper is to show that some combinatorial sequences, such as second-order Eulerian numbers and Eulerian numbers of type $B$, can be generated by context-free grammars.
Expressions are given for the exponential of a hermitian matrix, A. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA). They extend to any size matrix the previous results for the 2 X 2, 3 X 3, and 4 X 4…
Within the framework of stochastic Schroedinger equations, we show that the correspondence between statevector equations and ensemble equations is infinitely many to one, and we discuss the consequences. We also generalize the results of…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
We present a generalization of the well known Next-Closure algorithm working on semilattices. We prove the correctness of the algorithm and apply it on the computation of the intents of a formal context.
We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…
We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels…
In this paper we study odd unimodal and odd strongly unimodal sequences. We use $q$-series methods to find several fundamental generating functions. Employing the Euler--Maclaurin summation formula we obtain the asymptotic main term for…
The On-Line Encyclopedia Of Integer Sequences , that wonderful resource that most combinatorialists, and many other mathematicians and scientists, use at least once a day, is a treasure trove of mathematical information, and, one of its…