_{1}

^{*}

In this paper, we
will see that some *k* -Fibonacci sequences
are related to the classical Fibonacci sequence of such way that we can express the terms
of a *k* -Fibonacci sequence in
function of some terms of the classical Fibonacci sequence. And the formulas
will apply to any sequence of a certain set
of* k* -Fibonacci sequences. Thus find *k* -Fibonacci sequences relating to other-Fibonacci sequences when σ'_{k} is linearly dependent of

formations used in the well-known four-triangle longest-edge (4TLE) partition. This sequence generalizes the classical Fibonacci sequence [

For any positive real number

From this definition, the polynomial expression of the first

If

. Polynomial expression of the first k-Fibonacci numbers

quence is the classical Pell sequence

The characteristic equation of the recurrence equation of the definition of the

As particulars cases [

1) If

2) If

3) If

From now on, we will represent the classical Fibonacci numbers as

Binet identity takes the form [

Power

Proof. By induction. For

Obviously, the formulas found in [

In this section, we try to find the relationships that can exist between the values of

We can write this last equation as

because

Main problem is to solve the quadratic Diophantine equation

The positive characteristic root

For

Then,

In the same way, we can prove that

For

Proof. Taking into account both

It is worthy of note that Equation (2) is similar to the relationship between the elements of the

And

From two previous theorems, the

The values of the parameter of these sequences are

Next we present the first few values of the parameter

But these polynomials verify the relationship

where

The coefficients of these polynomials generate the triangle in

Last column is the sum by row of the coefficients, and it is a bisection of the classical Lucas sequence

If

diagonal plus 27 of the row 5 is the 77 of the row 6.

All the first diagonal sequences are listed in [

From this study, it is easy to find the values of “

Sequence

In this case, the triangle of coefficients is in

First diagonal sequences and the antidiagonal sequences are listed in OEIS.

Finally, for the values of

. Triangle of the coefficients of k_{n}

1 | 1 | 1 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 3 | 4 | |||||||||

3 | 1 | 5 | 5 | 11 | ||||||||

4 | 1 | 7 | 14 | 7 | 29 | |||||||

5 | 1 | 9 | 27 | 30 | 9 | 76 | ||||||

6 | 1 | 11 | 44 | 77 | 55 | 11 | 199 |

In this case, the triangle of the coefficients of the expressions of

Last column is the other bisection of the classical Fibonacci sequence.

The diagonal sequence

In this table, it is verified:

a)

b)

c) The diagonal sequences are listed in OEIS.

d) The elements of

Then we will apply the results to the

In this section we try to find the relations that could exist between the values of “

In this case, Equation (2) takes the form

The integer solutions of Equation

Proof. Applying Binnet Identity, and taking into account

. Triangle of the coefficients of b_{n}

1 | 1 | 1 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | 2 | ||||||||||

3 | 1 | 3 | 1 | 5 | |||||||||

4 | 1 | 5 | 6 | 1 | 13 | ||||||||

5 | 1 | 7 | 15 | 10 | 1 | 34 | |||||||

6 | 1 | 9 | 28 | 35 | 15 | 1 | 89 |

. Triangle of the coefficients of a_{n}

1 | 1 | 1 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 2 | 3 | |||||||||

3 | 1 | 4 | 3 | 8 | ||||||||

4 | 1 | 6 | 10 | 4 | 21 | |||||||

5 | 1 | 8 | 21 | 20 | 5 | 55 | ||||||

6 | 1 | 10 | 36 | 56 | 35 | 6 | 144 |

Consequently, the values of the parameter “

Integer solutions of this equation are expressed in

We will show some properties of the sequences of

The sequence of values of “

The sequence of values of “

The sequence of values of “

All these sequences verify the recurrence law given in Equation (2),

As a consequence of this situation, if we represent as

Applying Subsection 2.3 when

Consequently:

Repeating the previous process, we can solve the Diophantine equation

. Integer solutions of the Diophantine equation 5b^{2} – k^{2} = 4

k_{n} = L_{2n+1} | b_{n} = F_{2n+1} | a_{n} = F_{2n} | σ_{1,n} |
---|---|---|---|

1 | 1 | 0 | |

4 | 2 | 1 | |

11 | 5 | 3 | |

29 | 13 | 8 | |

76 | 34 | 21 |

The values obtained are showed in

We will show some properties of the sequences of

that

two consecutive terms of this sequence is the sequence

Much more interesting is the sequence obtained by dividing by 2:

All these sequences verify the recurrence law (2),

As in the preceding section, if we represent the sequence of values of “

Taking into account

Consequently:

Repeating the previous process, we can solve the Diophantine equation

The values obtained are showed in

. Integer solutions of the Diophantine equation 8b^{2} – k^{2} = 4

k_{n} = P_{2n} + P_{2n+2} | b_{n} = P_{2n+1} | a_{n} = P_{2n} | σ_{2,n} |
---|---|---|---|

2 | 1 | 0 | |

14 | 5 | 2 | |

82 | 29 | 12 | |

478 | 169 | 70 |

We will show some properties of the sequences of

All these sequences verify the recurrence law (Equation (2)),

The sequence

5.2. Relationships between the k–Fibonacci Sequences for

Taking into account

Consequently:

There are infinite

1) The relationship

Relationship between “

Diophantine equation:

2) Relationship between the positive characteristic root

3) Second sequence related to the

4) Two first values of “

5) Two first values of “

6) Recurrence law for the sequences

. Integer solutions of the Diophantine equation 13 b^{2} – k^{2} = 4

k_{n} | b_{n} = F_{3,2n+1} | a_{n} = F_{3,2n} | σ_{3,n} |
---|---|---|---|

3 | 1 | 0 | |

36 | 10 | 3 | |

393 | 109 | 33 | |

4287 | 1189 | 360 |

It is worthy of remark the fact the last sequence

the initial

4-Fibonacci sequence:

11-Fibonacci sequence:

29-Fibonacci sequence: