We can also generalize the expansions for phi mentioned in the last article

phi² = 1 + phi

phi³ = 1 + 2phi

phi⁴ = 2 + 3phi

phi⁵ = 3 + 5phi

phi⁶ = 5 + 8 phi,
 
 

to the 4-term series (i.e. whether we start with 0, 0, 0, 1 and each successive term is the sum of the four previous terms.
 
 

This leads to

0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 10671, 20569, 39648, 76424, 147312, 283953, 547337,…
 
 

Phi₄ =1.9275619…
 
 
 
 

Here expansions of (Phi₄)ⁿ where n > 4 can be expressed in terms of configurations of (Phi₄)⁴, (Phi₄)², Phi₄ and a constant. All the coefficients are based on the terms of the corresponding 4-term series.

(Phi₄)⁵ = 2(Phi₄)⁴ - 1; the (Phi₄)² and Phi₄ terms are not here involved.
 
 

(Phi₄)⁶ = 4(Phi₄)⁴ - Phi₄ - 2; the (Phi₄)² term is not here involved.
 
 

(Phi₄)⁷= 8(Phi₄)⁴ - (Phi₄)²- 2Phi₄ - 4;
 
 

(Phi₄)⁸= 15(Phi₄)⁴ - (Phi₄)²- 3Phi₄ - 7;
 
 

(Phi₄)⁹= 29(Phi₄)⁴ - 2(Phi₄)²- 6Phi₄ - 14;
 
 

(Phi₄)¹⁰= 56(Phi₄)⁴ - 4(Phi₄)²- 12Phi₄ - 27;
 
 

Now in the last example where n = 10 the coefficient of (Phi₄)⁴ = T_{n + 1;} the coefficient of (Phi₄)² = T_{n -3;} the constant term = T_{n + 1} - T_n; finally the coefficient of the Phi₄ term = T_{n + 1} - T_n - T_{n - 1}
 

So 

(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);
 
 

This means that for (Phi₄)¹¹, the coefficient of (Phi₄)⁴ = T₁₂ = 108_; the coefficient of (Phi₄)² = T₈= 8; the coefficient of Phi₄ = T₁₂ - T₁₁ - T₁₀ = 108 - 56 - 29 = 23; finally the constant term = T₁₂ - T₁₁ = 108 - 56 = 52;
 
 

Therefore
 
 

(Phi₄)¹¹ = 108(Phi₄)⁴ - 8(Phi₄)² - 23Phi₄ - 52;