The formulas we have been generating in previous articles for 2-term, 3-term, 4-term and 5-term relationships can be extended for any number of terms.

In the 2-term case,

Phiⁿ = T_n + (T_{n + 1}) phi

In the 3-term case,

(Phi₃)ⁿ = T_{n + 1} (phi₃)³ - T_{n - 2}(phi₃) - (T_{n + 1} - T_n)

In the 4-term case,

(Phi₄)ⁿ = T_{n + 1}(Phi₄)⁴ - T_{n - 3}(Phi₄)² - (T_{n + 1} - T_n - T_{n - 1})Phi₄ - (T_{n + 1} - T_n);
 
 

In the 5-term case,

(Phi₅)ⁿ = T_{n + 1} (Phi₅)⁵ - T_{n - 4}(Phi₅)³ - (T_{n + 1} - T_n - T_{n - 1} - T_{n - 1}) (Phi₅)²
- (T_{n + 1} - T_n - T_{n - 1}) (Phi₅) - (T_{n + 1} - T_n)
 
 

Therefore extending this pattern of terms,

(Phi₆)ⁿ = T_{n + 1} (Phi₆)⁶ - T_{n - 5}(Phi₆)⁴ - (T_{n + 1} - T_n - T_{n - 1} - T_{n - 1} - T_{n - 2}) (Phi₆)³
- (T_{n + 1} - T_n - T_{n - 1} - T_{n - 1}) (Phi₆)² - (T_{n + 1} - T_n - T_{n - 1}) (Phi₆) - (T_{n + 1} - T_n)
 
 

The 6-term sequence is

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 482, 966, 1916, 3800, 7537, 14949, 29650, 58818, 116670, 231424, 459048, 910559, 1806169,…

The ratio is 1.98358…
 
 

So when for example n = 12,

T_{n + 1} = 63

T_{n - 5} = 1

T_{n + 1} - T_n - T_{n - 1} - T_{n - 1} - T_{n - 2} = 63 - 32 - 16 - 8 - 4 = 3

T_{n + 1} - T_n - T_{n - 1} - T_{n - 1} = 63 - 32 - 16 - 8 = 7

T_{n + 1} - T_n - T_{n - 1} = 63 - 32 - 16 = 15

T_{n + 1} - T_n = 63 - 32 = 31
 
 

Therefore

(Phi₆)¹² = 63(Phi₆)⁶ - (Phi₆)⁴ - 3 (Phi₆)³ - 7 (Phi₆)² - 15 (Phi₆) - 31