Related papers: Symmetric Numerical Semigroups Generated by Fibona…
In this paper we study numerical semigroups generated by three elements. We give a characterization of pseudo-symmetric numerical semigroups. Also, we will give a simple algorithm to get all the pseudo-symmetric numerical semigroups with…
In this paper we compute the Frobenius number of certain {\em Fibonacci numerical semigroups}, that is, numerical semigroups generated by a set of Fibonacci numbers, in terms of Fibonacci numbers.
We consider symmetric (not complete intersection) numerical semigroups S_6, generated by a set of six positive integers {d_1,...,d_6}, gcd(d_1,...,d_6)=1, and derive inequalities for degrees of syzygies of such semigroups and find the lower…
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by…
We derive the polynomial representations for minimal relations of generating set of numerical semigroups R_n^k=<(n-1)^k,n^k,(n+1)^k>, k=2,3,4, n>2. We find also the polynomial representations for degrees of syzygies in the Hilbert series…
Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c,…
In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers.…
We compute the Frobenius number for numerical semigroups generated by the squares of three consecutive Fibonacci numbers. We achieve this by using and comparing three distinct algorithmic approaches: those developed by Ram\'irez Alfons\'in…
We consider symmetric (not complete intersection) numerical semigroups S_5, generated by five elements, and derive inequalities for degrees of syzygies of S_5 and find the lower bound F_5 for their Frobenius numbers. We study a special case…
The notion of almost symmetric numerical semigroup was given by V. Barucci and R. Fr\"oberg. We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for $H^*$ (the dual of $M$) to…
For a nonnegative integer $p$, we give explicit formulas for the $p$-Frobenius number and the $p$-genus of generalized Fibonacci numerical semigroups. Here, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose…
We study almost symmetric semigroups generated by odd integers. If the embedding dimension is four, we characterize when a symmetric semigroup that is not complete intersection or a pseudo-symmetric semigroup is generated by odd integers.…
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have…
We derive the lower bound for Frobenius number of symmetric (not complete intersection) semigroups generated by four elements.
We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our…
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup…
We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if $n_g$ is the number of numerical semigroups of genus $g$, we prove that $n_g$ tends to $S \phi^g$, where $\phi$ is the golden ratio,…
We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Ap\'ery set, as well as bounds on the elements of the Ap\'ery…
We establish a one-to-one correspondence between numerical semigroups of genus $g$ and almost symmetric numerical semigroups with Frobenius number $F$ and type $F-2g$, provided that $F$ is greater than $4g-1$.
This contribution presents all possible solutions to the Diophantine equations $F_k=L_mL_n$ and $L_k=F_mF_n$. To be clear, Fibonacci numbers that are the product of two arbitrary Lucas numbers and Lucas numbers that are the product of two…