Related papers: General Dynamics and Generation Mapping for Collat…

We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals…

We propose a procedure to reconstruct characteristic features of an unknown stochastic system from the long-time full counting statistics of some of the system's transitions that are monitored by a detector. The full counting statistics is…

Let $(t_n)_{n\ge0}$ be the well konwn $\pm1$ Thue-Morse sequence $$+1,-1,-1,+1,-1,+1,+1,-1,\cdots.$$ Since the 1982-1983 work of Coquet and Dekking, it is known that $\sum_{k<n}t_ke^\frac{2k\pi i}{3}$ is strongly related to the famous Koch…

Looped Transformers offer a promising alternative to purely feed-forward computation by iteratively refining latent representations, improving language modeling and reasoning. Yet recurrent architectures remain unstable to train, costly to…

Answering a question of Donald Knuth, we find the bivariate exponential generating function for "up-up-or-down-down'' permutations of odd length according to their last entry. An up-up-or-down-down permutation is a permutation $a_1a_2\cdots…

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that…

Let $H_{m}(z)$ be a sequence of polynomials whose generating function $\sum_{m=0}^{\infty}H_{m}(z)t^{m}$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^{2}$,…

Define the map $\mathsf{T}$ on the positive integers by $\mathsf{T}(m)=\frac{m}{2}$ if $m$ is even and by $\mathsf{T}(m)=\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\epsilon>0$, almost all…

We prove moderate deviations bounds for the lower tail of the number of odd cycles in a $\calG(n, m)$ random graph. We show that the probability of decreasing triangle density by $t^3$, is $\exp(-\Theta(n^2 t^2))$ whenever $n^{-3/4} \ll t^3…

Only one three-term recurrence relation, namely, $W_{r}=2W_{r-1}-W_{r-4}$, is known for the generalized Tribonacci numbers, $W_r$, $r\in\mathbb{Z}$, defined by $W_{r}=W_{r-1}+W_{r-2}+W_{r-3}$ and \mbox{$W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}$},…

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel $\sim 1/t^{1+\theta}$ with $0 \leq \theta <1$. In the presence of an epidemic-like process of…

Collatz conjecture is generalized to $3n+3^k$ ($k\in N$). Operating as usual, every sequence seems to reach $3^k$ and end up in the loop $3^k, 4.3^k, 2.3^k,3^k$. The usual $3n+1$ conjecture is recovered for $k=0$. For $k>0$, we noticed the…

We present a comprehensive structural analysis of the Collatz conjecture through ~1014 computational experiments yielding 630 formal results. By systematically deploying 29 distinct mathematical paradigms--including transfer operator…

Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the…

A look-and-say sequence is obtained iteratively by reading off the digits of the current value, grouping identical digits together: starting with 1, the sequence reads: 1, 11, 21, 1211, 111221, 312211, etc. (OEIS A005150). Starting with any…

There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let $G$ be a graph with minimum degree at least $k+1$. We prove that if $G$ is…

Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the…

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic…

Two evolution models based on the generalized Collatz operator are introduced. These models are characterized by coefficients $\alpha$ and $\beta$ in the Collatz dynamics, and are suitably defined. Here, $\alpha=\beta=1$, and $\alpha=3$,…

We derive the recurrence relation of irreducible tensor operator for O(4) in using the Wigner-Eckart theorem. The physical process like radiative transitions in atomic physics, nuclear transitions between excited nuclear states can be…