Consider the recurrence relation

Under what conditions on x and y are the constants in the general solution to the recurrence integers?

Assume this is correct:

then

I have figured out by trial and error that

x and y can be any integer that when modded with 7 is congruent to 0.

or

If x and y are divided by 7 and they leave no remainder,

then that will produce the constants and

Can someone please explain to me why this happens?