ALGEBRAIC EXPRESSION FOR PHI3

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 + (51/2)/2

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

= 1/3 + a cube root expression