EXTENSION OF INTERESTING FIBONACCI RESULT TO 5-TERM SEQUENCE



 

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 = Tn + 1 (Phi5)5 - Tn - 4(Phi5)3 - (Tn + 1 - Tn - Tn - 1 - Tn - 1) (Phi5)2
- (Tn + 1 - Tn - Tn - 1) (Phi5) - (Tn + 1 - Tn)
 
 

Therefore when n = 11,

Tn + 1 = 61,

Tn - 4 = 2;

(Tn + 1 - Tn - Tn - 1 - Tn - 1) = 61 - 31 - 16 - 8 = 6;

Tn + 1 - Tn - Tn - 1 = 61 - 31 - 16 = 14;

Tn + 1 - Tn = 61 - 31 = 30
 
 

Thus

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