The procedure adopted in the article can be summed up through a well-ordered series of fascinating connections between phi and the respective terms of the Lucas series
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 621, 943, ….
Phi - 1/Phi = 1 (the first term of Lucas Series)
Phi2 + 1/Phi2 = 3 (the second term of Lucas series)
Phi3 - 1/Phi3 = 4 (the third term of Lucas Series)
Phi4 + 1/Phi4 = 7 (the fourth term of Lucas Series)
Phi5 - 1/Phi5 = 11 (the fifth term of Lucas Series)
So in general format
Phin (+ or -) 1/Phin = Tn (where Tn represents the nth term of the Lucas Series).
When n is even we add the respective values of Phi.
When n is odd, we subtract the respective values of Phi.
Thus for example T11 = Phi11 - 1/Phi11 = 199 (which is the 11th term in the series).