Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows:
  
  



$a=4k-2,$ $y\_0 = 1,$ $y\_1 = a + 1,y\_n = ay\_{n-1} - y\_{n-2}$
  
  



$b = 4k + 2,$ $ x\_0 = 1,$ $x\_1 = b - 1,$ $x\_n = bx\_{n-1} - x\_{n-2}$


  
  

For example, with $k = 2$ we get 


$y\_j = 7, 41, 239, 1393, \ldots$


$x\_j = 9, 89, 881, 8721, \ldots$


A simple question arises, as to whether there exist $\{k, i, j\}$ such that $X(k,i) = Y(k,j)$?


This might well be an open question, and perhaps inappropriate here, but I have trawled the web for many hours and have found no evidence that anybody has even considered it.


Computational experiments suggest that in fact an even stronger result is possible, ie. that there are no $\{k\_1, k\_2, i>1, j>1\}$ with $X(k\_1,i) = Y(k\_2,j)$.


In other words, with the exception of $x\_1, y\_1$ which can be any odd number > 7, all values generated by these sequences appear to be unique.


Any suggestions as to a way to attack this question would be greatly appreciated!


Update: There are explicit proofs that for $k = 2, 3$ there can be no $X(k,i) = Y(k,j)$, so we can restrict the question to $k > 3$. Sadly these proofs are not extendable to other *k*



Uniqueness of values in recurrence relations
You may consider the paper:


B. Ibrahimpasic, A parametric family of quartic Thue inequalities. 
Bull. Malays. Math. Sci. Soc. (2) 34 (2011), no. 2, 215–230, 
available at <http://www.emis.de/journals/BMMSS/vol34_2_2.html> 


It seems that Theorem 3.1, with c=2k, answers your question.


