The plus-minus sequence

Jelly, 5 bytes

ṄI;Sß

This is a recursive approach. Due to tail call optimization, the only limit is the ability to fit both integers into memory. Output is one list per line.

Try it online!

How it works

ṄI;Sß  Main link. Argument: [b[n], a[n]] (n = 0 for original input)

Ṅ      Print [b[n], a[n]] to STDOUT.
 I     Compute the increments of the list, i.e., [a[n] - [b[n]].
   S   Compute the sum of the list, i.e., b[n] + a[n].
  ;    Concatenate the results to the left and to the right.
    ß  Recursively call the main link.

Python 2, 31 bytes

def f(a,b):print a,b;f(a+b,a-b)

Prints forever. Well, eventually you exceed the recursion limit, but that's a system limitation.

MATL, 10 bytes

`tDtswPdhT

This version will output an infinite number of elements in the plus-minus sequence.

Try it Online! (stop it after running due to infinite loop)

Explanation

    % Implicitly grab input as a two-element array [a,b]
`   % do...while loop
tD  % Duplicate and display the top of the stack
ts  % Duplicate [a,b] and add them together
w   % Swap the top two elements on the stack
P   % Swap the order of b and a in preparation for diff
d   % Compute the difference between b and a
h   % Horizontally concatenate [a+b, a-b]
T   % Explicit TRUE to make it an infinite loop
    % Implicit end of the do...while loop