EXTENSION OF INTERESTING FIBONACCI RESULT TO 3-TERM SEQUENCE

In the two-term Fibonacci sequence, higher powers of phi can be expressed in terms of simple combinations of phi (which themselves are related in orderly fashion to the Fibonacci sequence).

Thus phi2 = 1 + phi

phi3 = 1 + 2phi

phi4 = 2 + 3phi

phi5 = 3 + 5phi

phi6 = 5 + 8 phi

So the coefficients of both terms increase in accordance with the Fibonacci sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34,…..

In this last case for example where phi is raised to the power of 6, 5 is the 6th term and 8 the 7th term.

Therefore in general

Phin = Tn + (Tn + 1) phi

Equally fascinating - though more complex - formulations can be obtained for the more generalized n-term Fibonacci sequences.

Once again in the three-term case (starting with 0. 0, 1) and continually combining the last 3 term to obtain the next term, we obtain the followibg sequence

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927,….

The ratio here of k/(k - 1), where k represents the value of the kth term in the sequence = phi3 = 1.839286755…

Now (phi3)n (where n >3) can be expressed through orderly combinations of (phi3)3 , phi3 and a constant (which involve the 3-term sequence in an ordered fashion).

Thus

(phi3)4 = 2(phi3)3 - 0(phi3) - 1, i.e. 2(phi3)4 -1

(phi3)5 = 4(phi3)3 - phi3 - 2

(phi3)6 = 7(phi3)3 - phi3 - 3

(phi3)7 = 13(phi3)3 - 2(phi3) - 5

(phi3)8 = 24(phi3)3 - 4(phi3) - 11

(phi3)9 = 44(phi3)3 - 7(phi3) - 20

(phi3)10 = 81(phi3)3 - 13(phi3) - 37

For example, in this last case where n = 10, the coefficient of the Ist term (81) is the 11th term (i.e. Tn + 1) in the sequence (starting 0. 0, 1). The coefficient of the 2nd term (13) is the 8th (i.e. Tn - 2). Finally the last term (37) is the difference of Tn + 1 - Tn.

Therefore in general terms

(phi3)n = Tn + 1 (phi3)3 - Tn - 2(phi3) - (Tn + 1 - Tn)

So for example when n= 14,

Tn + 1 = 927,  Tn - 2 = 149 and Tn + 1 - Tn = 927 - 504 = 423.

Therefore

(phi3)14 = 927(phi3)3 - 149(phi3) - 423