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

phi^{2 }= 1 + phi

phi^{3 }= 1 + 2phi

phi^{4 }= 2 + 3phi

phi^{5 }= 3 + 5phi

phi^{6 }= 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,…

Phi_{4 }=1.9275619…

Here expansions of (

Phican be expressed in terms of configurations of (_{4})^{n }where n > 4Phi(_{4})^{4},^{ }Phiand a constant. All the coefficients are based on the terms of the corresponding 4-term_{4})^{2},^{ }Phi_{4}series.

(Phi(_{4})^{5}_{ }= 2Phithe_{4})^{4 }- 1;(^{ }Phiterms_{4})^{2}and^{ }Phi_{4 }are not here involved._{ }

(Phi(_{4})^{6}_{ }= 4Phithe_{4})^{4 }- Phi_{4}- 2;(^{ }Phiterm is not here involved._{4})^{2}

(Phi(_{4})^{7}= 8Phi(_{4})^{4 }-Phi_{4})^{2}- 2Phi_{4}- 4;

(Phi(_{4})^{8}= 15Phi(_{4})^{4 }-Phi_{4})^{2}- 3Phi_{4}- 7;

(Phi(_{4})^{9}= 29Phi(_{4})^{4 }- 2Phi_{4})^{2}- 6Phi_{4}- 14;

(Phi(_{4})^{10}= 56Phi(_{4})^{4 }- 4Phi_{4})^{2}- 12Phi_{4}- 27;

Now in the last example

where n = 10the coefficient of(Phithe coefficient of (_{4})^{4 }= T_{n + 1; }Phithe constant term =_{4})^{2 }= T_{n -3; }Tfinally the coefficient of the_{n + 1 }- T_{n; }Phiterm_{4 }= T_{n + 1 }- T_{n }- T_{n - 1}_{}_{}So

_{}(

Phi_{4})^{n }= T(_{n + 1}Phi(_{4})^{4 }- T_{n - 3}Phi_{4})^{2 }- (T_{n + 1 }- T_{n }- T_{n - 1})Phi_{4}- (T_{n + 1 }- T_{n});

^{}This means that for

(Phithe coefficient_{4})^{11},of(Phithe coefficient_{4})^{4 }= T_{12 }= 108_{; }of(Phithe coefficient_{4})^{2 }= T_{8}= 8;ofPhiinally the constant term =_{4}^{ }= T_{12 }- T_{11 }- T_{10 }= 108 - 56 - 29 = 23; fT_{12 }- T_{11 }= 108 - 56 = 52;

Therefore

(Phi(_{4})^{11 }= 108Phi(_{4})^{4 }- 8Phi_{4})^{2 }- 23Phi_{4}- 52;