Related papers: Study of Meta-Fibonacci Integer Sequences by Conti…

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform…

We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique.…

The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the…

In a recent insightful article, Helmut Prodinger uses sophisticated complex analysis, with residues, to derive convolution identities for Fibonacci, Tribonacci, and k-bonacci numbers. Here we use a naive, "experimental mathematics" (yet…

Inspired by the surprising relationship (due to A. Bird) between Schreier sets and the Fibonacci sequence, we introduce Schreier multisets and connect these multisets with the $s$-step Fibonacci sequences, defined, for each $s\geqslant 2$,…

Golomb's sequence is the unique nondecreasing sequence of positive integers in which each $n$ appears exactly $a(n)$ times. It satisfies the global self-referential rule \[ a\bigl(a(n)+a(n-1)+\cdots+a(1)\bigr)=n, \] grows smoothly like a…

The normal sequences NS(n) and near-normal sequences NN(n) play an important role in the construction of orthogonal designs and Hadamard matrices. They can be identified with certain base sequences (A;B;C;D), where A and B have length n+1…

One-parameter Darboux deformations are effected for the simple ODE satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with…

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,...,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In 2005, Noe and Post conjectured…

Consider "lagged" Fibonacci sequences $a(n) = a(n-1)+a(\lfloor n/k\rfloor)$ for $k > 1$. We show that $\lim_{n\to\infty} a(kn)/a(n)\cdot\ln n/n = k\ln k$ and we demonstrate the slow numerical convergence to this limit and how to deal with…

In this paper, we investigate the existence of $C^n$, $n\in \mathbb{N}^+$, solutions for a class of second-order iterative functional equations involving iterates of the unknown function and a nonlinear term. Applying the Fiber Contraction…

For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. The it is well known that only finitely many positive integers…

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\…

We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of…

Two $k$-ary Fibonacci recurrences are $a_k(n) = a_k(n-1) + k \cdot a_k(n-2)$ and $b_k(n) = k \cdot b_k(n-1) + b_k(n-2)$. We provide a simple proof that $a_k(n)$ is the number of $k$-regular words over $[n] = \{1,2,\ldots,n\}$ that avoid…

We derive the double recurrence $e_n = \frac{1}{2}(a_{n-1}+5b_{n-1}); f_{n} = \frac{1}{2}(a_{n-1}+b_{n-1})$ with $e_0=2;f_0=0$ for the Fibonacci numbers, leading to an extremely simple and fast implementation. Though the recurrence is…

We study the 2-adic behavior of the number of domino tilings of a 2n-by-2n square as nvaries. It was previously known that this number was of the form 2^n f(n)^2, where f(n) is an odd, positive integer. We show that the function f is…

Let $\UT_n(\FF_q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of $\UT_{n+1}(\FF_q)$ are indexed by lattice paths from the origin to the…