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.

 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 1 1 1 2 3 5 8 13 21 34 55 89 144 233 2 1 1 1 2 3 4 6 9 13 19 28 41 60 3 1 1 1 1 2 3 4 5 7 10 14 19 28 4 1 1 1 1 1 2 3 4 5 7 9 12 16

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 ð > 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.

 n 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 F1(n) 21 13 8 5 3 2 1 1 0 1 -1 2 -3 5 -8 13 -21 34 F2(n) 9 6 4 3 2 1 1 1 0 0 1 0 -1 1 1 -2 0 2 F3(n) 5 4 3 2 1 1 1 1 0 0 0 1 0 0 -1 1 0 1 F4(n) 4 3 2 1 1 1 1 1 0 0 0 0 1 0 0 0 -1 1 F5(n) 3 2 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 -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!