EXTENSION OF INTERESTING FIBONACCI RESULT TO 4-TERM SEQUENCE

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.

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 = Tn + 1; the coefficient of (Phi4)2 = Tn -3; the constant term = Tn + 1 - Tn; finally the coefficient of the Phi4 term = Tn + 1 - Tn - Tn - 1

So

(Phi4)
= Tn + 1(Phi4)4 - Tn - 3(Phi4)2 - (Tn + 1 - Tn - Tn - 1)Phi4 - (Tn + 1 - Tn);

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;