New results and conjectures on 2-partitions of multisets.

The interplay between integer sequences and partitions has led to numerous interesting results, with implications in generating functions, integral formulae, or combinatorics. An illustrative example is the number of solutions at level n to the signum equation. Denoted by S(n), this represents the number of ways of choosing + and - such that ±1±2±3±···±n = 0 (see A063865 in OEIS). The Andrica-Tomescu conjecture regarding the asymptotic behaviour of S(n) was solved affirmatively in 2013, and new conjectures were formulated since then. In this paper we present recurrence formulae, generating functions and integral formulae for the number of ordered 2-partitions of the multiset M having equal sums. Certain related integer sequences not currently indexed in the OEIS are then presented. Finally, we formulate conjectures regarding the unimodality, distribution and asymptotic behaviour of these sequences.N/

On some results concerning the polygonal polynomials.

In this paper we define the $n$th polygonal polynomial $P_n(z) = (z-1)(z^2-1)\cdots(z^n-1)$ and we investigate recurrence relations and exact integral formulae for the coefficients of $P_n(z)$ and for those of the Mahonian polynomials $Q_n(z)=(z+1)(z^2+z+1)\cdots(z^{n-1}+\cdots+z+1)$. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.O. Bagdasar's research was supported by a grant of the Romanian National Authority for Research and Innovation, CNCS/CCCDI UEFISCDI, project number PN-III-P2-2.1-BG-2016-0333, within PNCDI III

Equations with Solution in Terms of Fibonacci and Lucas Sequences

Abstract The main results characterize the equations (2.1) and (2.10) whose solutions are linear combinations with rational coefficients of at most two terms of classical Fibonacci and Lucas sequences

On Generalized Lucas Pseudoprimality of Level k

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k− and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences.N/

On some new arithmetic properties of the generalized Lucas sequences

Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.N/

Remarks on a family of complex polynomials

Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1, . . . , mn, . . . , and on a sequence of complex numbers z1, . . . , zn, . . . of modulus one. We investigate some particular instances such as: extended cyclotomic, extended polygonal-type, and multinomial polynomials, for which we obtain formulae for the coefficients. Some novel related integer sequences are also derived.N/

Some remarks on 3-partitions of multisets.

Partitions play an important role in numerous combinatorial optimization problems. Here we introduce the number of ordered 3-partitions of a multiset M having equal sums denoted by S(m1, ..., mn; α1, ..., αn), for which we find the generating function and give a useful integral formula. Some recurrence formulae are then established and new integer sequences are added to OEIS, which are related to the number of solutions for the 3-signum equation.O. Bagdasar’s research was supported by a grant of the Roma- nian National Authority for Research and Innovation, CNCS/CCCDI UEFISCDI, project number PN-III-P2-2.1-PED-2016-1835, within PNCDI III

A new formula for the coefficients of Gaussian polynomials

We deduce exact integral formulae for the coefficients of Gaussian, multinomial and Catalan polynomials. The method used by the authors in the papers [2, 3, 4] to prove some new results concerning cyclotomic and polygonal polynomials, as well as some of their extensions is applied.O. Bagdasar’s research was supported by a grant of the Romanian National Authority for Research and Innovation, CNCS/CCCDI UEFISCDI, project number PN-III-P2-2.1-BG-2016-0333, within PNCDI III