The five-term series is
 

0, 0, 0, 0, 1, 1 , 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577…
 
 

The ratio is 1.96594824 (approx) = Phi₅
 
 

(Phi₅)⁶ = 2(Phi₅)⁵ - 1
 
 

(Phi₅)⁷ = 4(Phi₅)⁵ - Phi₅ - 2
 
 

(Phi₅)⁸ = 8(Phi₅)⁵ - (Phi₅)² - 2(Phi₅) - 4
 
 

(Phi₅)⁹ = 16(Phi₅)⁵- (Phi₅)³ - 2(Phi₅)² - 4(Phi₅) - 8
 
 

(Phi₅)¹⁰ = 31(Phi₅)⁵- (Phi₅)³ - 3(Phi₅)² - 7(Phi₅) - 15
 

So in general terms,

(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 when n = 11,

T_{n + 1} = 61,

T_{n - 4} = 2;

(T_{n + 1} - T_n - T_{n - 1} - T_{n - 1}) = 61 - 31 - 16 - 8 = 6;

T_{n + 1} - T_n - T_{n - 1} = 61 - 31 - 16 = 14;

T_{n + 1} - T_n = 61 - 31 = 30
 
 

Thus

(Phi₅)¹¹ = 61(Phi₅)⁵- 2(Phi₅)³ - 6(Phi₅)² - 14(Phi₅) - 30