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 = T_n (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 11^th term in the series).