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

Mathematica Bohemica (2012)

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

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## Abstract

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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 $F\cap T=\left\{0,1,2,13\right\}$.

## How to cite

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Marques, Diego. "On the intersection of two distinct $k$-generalized Fibonacci sequences." Mathematica Bohemica 137.4 (2012): 403-413. <http://eudml.org/doc/247189>.

@article{Marques2012,
abstract = {Let $k\ge 2$ and define $F^\{(k)\}:=(F_n^\{(k)\})_\{n\ge 0\}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^\{(k)\}=F_\{n-1\}^\{(k)\}+F_\{n-2\}^\{(k)\}+\cdots + F_\{n-k\}^\{(k)\}$, with initial conditions $0,0,\dots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^\{(k)\}=1$. The sequences $F:=F^\{(2)\}$ and $T:=F^\{(3)\}$ 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 $F_n^\{(k)\}=F_m^\{(\ell )\}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^\{(k)\}\cap F^\{(m)\}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\lbrace 0,1,2,13\rbrace$.},
author = {Marques, Diego},
journal = {Mathematica Bohemica},
keywords = {$k$-generalized Fibonacci numbers; linear forms in logarithms; reduction method; -generalized Fibonacci numbers; Tribonacci numbers; linear forms in logarithms; reduction method},
language = {eng},
number = {4},
pages = {403-413},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the intersection of two distinct $k$-generalized Fibonacci sequences},
url = {http://eudml.org/doc/247189},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Marques, Diego
TI - On the intersection of two distinct $k$-generalized Fibonacci sequences
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 4
SP - 403
EP - 413
AB - Let $k\ge 2$ and define $F^{(k)}:=(F_n^{(k)})_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{(k)}=F_{n-1}^{(k)}+F_{n-2}^{(k)}+\cdots + F_{n-k}^{(k)}$, with initial conditions $0,0,\dots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{(k)}=1$. The sequences $F:=F^{(2)}$ and $T:=F^{(3)}$ 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 $F_n^{(k)}=F_m^{(\ell )}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{(k)}\cap F^{(m)}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\lbrace 0,1,2,13\rbrace$.
LA - eng
KW - $k$-generalized Fibonacci numbers; linear forms in logarithms; reduction method; -generalized Fibonacci numbers; Tribonacci numbers; linear forms in logarithms; reduction method
UR - http://eudml.org/doc/247189
ER -

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