The values of x are the generalized Pell numbers P(n,5,5)
(OEIS A141448) beginning with 1: (1, 2, 5, 13, 34,...) and their associative negative equivalent (1, 2, 5, 13, 34,...).
The values of y are the bisection of the Fibonacci sequence F(2n) (OEIS A001906): (0, 1, 3, 8, 21, 55,...) and their associative negative equivalent (0, 1, 3, 8, 21, 55, ....).
There exists a twotoone correspondence between x and y, where each value maps to two values of the other. This can be observed by looking at the set of numbers (x,y):
Below I present the pairs (x, y) as (x, (y
_{1}, y
_{2})) where
(x, (y
_{1}, y
_{2}) = (x,y
_{1}) and (x, y
_{2}). I assigned y
_{1} as the "negative" values of y, and y
_{2} as the "positive" values of y. The "positive" values of y are "offset one position upwards" respective to the "negative" y values). The values of x, as does the corresponding Pell number P(n,5,5) sequence, continues toward infinity, but in this case also to negative infinity. y
_{1} continues toward negative infinity and y
_{2} continues toward infinity as does its corresponding sequence of the bisection of Fibonacci numbers. Thus there are an infinite number of integer solutions.
...
(34, (55, 21)),
(13, (21, 8)),
( 5, ( 8, 3)),
( 2, ( 3, 1)),
( 1, ( 1, 0)),
( 1, ( 0, 1)),
( 2, ( 1, 3)),
( 5, ( 3, 8)),
( 13, ( 8, 21)),
( 34, (21, 55)),
...
Edited on December 18, 2014, 11:21 am

Posted by Dej Mar
on 20141217 10:26:40 