Once again the 3-term Fibonacci type sequence is

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, ….

(This is obtained by successively combining the last 3-terms to obtain the next term in the sequence to obtain the next term).

The common ratio of k/(k + 1) where k represents the kth term approximates to 1.839286755…

I refer to this ratio (of the 3-term sequence) as Phi₃
 
 

As we have seen Phi₃ represents the real (positive- valued) solution for x to the cubic equation

x³ - x² - x - 1 = 0;

The solution to this equation can be expressed as follows:

x = Phi₃ = 1/3 + {19/27 + (11/27)^1/2}^1/3 + {19/27 - (11/27)^1/2}^1/3

There would appear to be a link here with the actual terms in the original sequence

To do this we express we multiply the fractional terms inside the bracket by 3/3 so that

x = Phi₃ = 1/3 + {57/81 + (33/81)^1/2}^1/3 + {57/81 - (33/81)^1/2}^1/3

The denominator of the terms (inside both sets of brackets) is 81 = T₁₁ in the sequence.

Also the difference of numerators (inside both sets of brackets) = 57 - 33 = 24 = T₉;

In the 2-term case for the Fibonacci Sequence, Phi₂ represents the positive solution for x to the quadratic equation

x² - x - 1 = 0;

x = Phi₂ (= Phi) = 1/2 + (5^1/2)/2
 
 

Thus whereas the two-term ratio = 1/2 + a square root expression, the three-term ratio

= 1/3 + a cube root expression