RECENT ADDENDUM
 
 
 
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, .

Thus

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).