# On the intersection of two distinct $k$-generalized Fibonacci sequences

Mathematica Bohemica (2012)

• Volume: 137, Issue: 4, page 403-413
• ISSN: 0862-7959

top Access to full text Full (PDF)

## Abstract

top
Let $k\ge 2$ and define ${F}^{\left(k\right)}:={\left({F}_{n}^{\left(k\right)}\right)}_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation ${F}_{n}^{\left(k\right)}={F}_{n-1}^{\left(k\right)}+{F}_{n-2}^{\left(k\right)}+\cdots +{F}_{n-k}^{\left(k\right)}$, with initial conditions $0,0,\cdots ,0,1$ ($k$ terms) and such that the first nonzero term is ${F}_{1}^{\left(k\right)}=1$. The sequences $F:={F}^{\left(2\right)}$ and $T:={F}^{\left(3\right)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation ${F}_{n}^{\left(k\right)}={F}_{m}^{\left(\ell \right)}$. In this note, we use transcendental tools to provide a general method for finding the intersections ${F}^{\left(k\right)}\cap {F}^{\left(m\right)}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\left\{0,1,2,13\right\}$.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.