One more about the generalized Fibonacci numbers

So called generalized Fibonacci numbers resulting from Pascal Triangle play the important role in our Harmony Mathematics. For the given non-negative integer p (p = 0, 1, 2, 3, ...) the generalized Fibonacci numbers called the p-Fibonacci numbers are given with the following recurrent formula:

Fp(n) = Fp(n - 1) + Fp(n-p-1) with n > p + 1;(1)
Fp(1) = Fp(2) = ... = Fp(p) = Fp(p+1) = 1(2)

Note that the p-Fibonacci numbers given with (1), (2) are the wide generalization of the classical Fibonacci numbers being the partial case of the p-Fibonacci numbers for p = 1.

The recurrent formula (1) at the initial conditions (2) generates the infinite number of the numerical series, which members are called the p-Fibonacci numbers (see Table 1).

Table 1.

12345678910111213
01248163264128256512102420484096
11123581321345589144233
2111234691319284160
311112345710141928
4111112345791216

For the case p = 0 the series of the p-Fibonacci numbers is reduced to the classical "binary" series. Let's consider now the p-Fibonacci numbers for the cases > 0 and extend them to the negative values of the argument of n. For that we will use the recurrent relation (1) and the initial condition (2) for calculation of the p-Fibonacci numbers Fp(0), Fp(-1), Fp(-2), ..., Fp(-p), ..., Fp(-2p + 1). Representing the p-Fibonacci number of Fp(p + 1) in the form of (1) we get:

Fp(p+1) = Fp(p) + Fp(0).(3)

Since according to (2) Fp(p) = Fp(p + 1) = 1 it follows from (3) that Fp(0) = 0.

Continuing this process, that is, representing the p-Fibonacci numbers Fp(p), Fp(p - 1), ..., Fp(2) in the form (1) we get:

Fp(0) = Fp(-1) = Fp(-2) = ... = Fp(-p + 1) = 0.(4)

Let's represent the number Fp(1) in the form:

Fp(1) = Fp(0) + Fp(-p).(5)

Since Fp(1) = 1 and Fp(0) = 0 it follows from (5) that

Fp(-p) = 1.(6)

Representing the p-Fibonacci numbers Fp(0), Fp(-1), ..., Fp(-p + 1) in the form (1) we can find that

Fp(-p - 1) = Fp(-p - 2) = ... = Fp(-2p + 1) = 0.(7)

Continuing this process we can get all values of the p-Fibonacci numbers Fp(n) for the negative values of n (see Table 2).

Table 2.

n876543210-1-2-3-4-5-6-7-8-9
F1(n)211385321101-12-35-813-2134
F2(n)964321110010-111-202
F3(n)54321111000100-1101
F4(n)4321111100001000-11
F5(n)32111111000001000-1

And now we have all reasons to introduce one more complicated concept, the concept of the generalized Fibonacci matrix called Qp-matrix. Follows us!