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This note is devoted to study the recurrent numerical sequence defined by: $a_0 = 0$, $a_n = \frac{n}{2} a_{n - 1} + (n - 1)!$ ($\forall n \geq 1$). Although, it is immediate that ${(a_n)}_n$ is constituted of rational numbers with…

We prove that Ma\~n\'e's conjecture, as stated in {\em Lagrangian flows: the dynamics of globally minimizing orbits}, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 141--153, contains another conjecture of Ma\~n\'e, stated in {\em Generic…

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of…

We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…

In 1998, Lin presented a conjecture on a class of ternary sequences with ideal 2-level autocorrelation in his Ph.D thesis. Those sequences have a very simple structure, i.e., their trace representation has two trace monomial terms. In this…

We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that…

Jacobi said "man muss immer umkehren". And indeed it takes a genius like Michael Somos to take a specific non-linear recurrence, like a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4), subject to a(1)=1, a(2)=1, a(3)=1, a(4)=1, and observe that surprise,…

Mason's Conjecture asserts that for an $m$--element rank $r$ matroid $\M$ the sequence $(I_k/\binom{m}{k}: 0\leq k\leq r)$ is logarithmically concave, in which $I_k$ is the number of independent $k$--sets of $\M$. A related conjecture in…

I study the recurrence D(n)= D(D(n-1))+D(n-1-D(n-2)), D(1)=D(2)=1. Its definition has some similarity to that of Conway's sequence defined through a(n)= a(a(n-1))+a(n-a(n-1)), a(1)=a(2)=1. However, in contradistinction to the completely…

Let $n>1$ be an odd integer. For any primitive $n$-th root $\zeta$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum_{\tau\in…

The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem. Take any positive integer $ n $. If $ n $ is even then divide it…

In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…

In this paper we present a new proof of the following 2010 result of Dubickas, Novikas, and Siurys: Let $(a,b)\in \mathbb{Z}^2$ and let $(x_n)_{n\ge 0}$ be the sequence defined by some initial values $x_0$ and $x_1$ and the second order…

A multipermutation with $k$ copies each of $1\ldots n$ is Carlitz if neighbours are different. We enumerate these objects for $k=2,3,4$ and derive recurrences. In particular, we prove and improve a conjectured recurrence for $k=3$, stated…

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the…

As a direct continuation of K. Zwart, arXiv:2304.02648, which is built on the work of M. M\"uger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for $Sp(N)$ and $G_2$ to a conjecture involving functions…

We introduce and analyze a three-parameter family of self-referential integer sequences $S(x,y,z)$: starting from $a(1)=x$, each term advances by $y$ when the index $k$ has already appeared as a value and by $z$ otherwise. This simple rule…

In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(\pi…

I want to show one possibility to proof the Collatz conjecture, also called 3n+1 conjecture, for any natural number N. For this, I limit my analysis on the direct odd follower of every natural odd number and show the connections between the…

The Eisenbud-Green-Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring $K[x_1,\,\ldots,\,x_n]$ over a field $K$ that contains a regular sequence $f_1,\,\ldots,\, f_n$ with degrees $a_i$, $i=1,\,\ldots,\,n$ has the…