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!