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 6^th term and 8 the 7^th term.

Therefore in general

Phiⁿ = T_n + (T_{n + 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. T_{n + 1}) in the sequence (starting 0. 0, 1). The coefficient of the 2^nd term (13) is the 8^th (i.e. T_{n - 2}). Finally the last term (37) is the difference of T_{n + 1} - T_n.
 
 

Therefore in general terms

(phi₃)ⁿ = T_{n + 1} (phi₃)³ - T_{n - 2}(phi₃) - (T_{n + 1} - T_n)
 
 

So for example when n= 14,

T_{n + 1} = 927,  T_{n - 2} = 149 and T_{n + 1} - T_n = 927 - 504 = 423.
 

Therefore

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