Generalized Fibonacci numbers

We have considered in the presiding page of our Museum an important mathematical object called Pascal Triangle. And now we will do some transformations on the Pascal Triangle.

Let us shift each row of the Pascal triangle in one column to the right about the preceding row. As the result of such transformation we get the following number array called the 1-Pascal triangle:

1-Pascal triangle

 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 1 3 6 10 15 21 28 36 1 4 10 20 35 56 1 5 15 35 1 6 1 1 2 3 5 8 13 21 34 55 89 144

It is easy to prove that the sum of the binomial coefficients in the n-th column of the 1-Pascal triangle is equal to the Fibonacci number Fn+1. This means that if we move along the bottom row of the 1-Pascal Triangle starting since 0-th column we get the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, ...!

Now, if we shift each row of the initial Pascal Triangle in the p columns to the right about the preceding row (p = 0, 1, 2, 3, ... ), we get number array called the p-Pascal triangle. It is clear that 0-Pascal Triangle is the initial Pascal Triangle. For example, the p-Pascal triangles corresponding to p = 2 and p = 3 have the following forms respectively:

2-Pascal triangle

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 1 3 6 10 15 21 28 1 4 10 20 1 1 1 1 2 3 4 6 9 13 19 28 41 60

3-Pascal triangle

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 3 6 10 15 1 1 1 1 1 2 3 4 5 7 10 14 19 26

Summing the binomial coefficients in the columns of the 2- and 3-Pascal triangles we get two new numerical sequences having the following properties. The n-th member of the sequence

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, ...

starting from n = 4 is equal to the sum of the (n - 1)-th and (n - 3)-th members, but the n-th member of the sequence

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, ...

starting from n = 5 is equal to the sum of the (n - 1)-th and (n - 4)-th members.

These examples give us a right to introduce the following definition.

Definition.
For the given p = 0, 1, 2, 3, ... the numerical sequences Fp(n) given with the following recurrent correlation
 Fp(n) = Fp(n-1) + Fp(n-p-1)   c   n>p+1; (1) Fp(1) = Fp(2) = ... = Fp(p+1) = 1 (2)

is called the p-Fibonacci numbers. We can see from the p-Pascal triangle that the sum of the binomial coefficients in the nth column of the p-Pascal triangle is equal to the p-Fibonacci number Fp(n+1).

Thus, we have discovered an infinite set of the number sequences consisting of the p-Fibonacci numbers (p = 0, 1, 2, 3, ...). These number sequences comprise the binary sequence (for p = 0) and the classical Fibonacci numbers (for p = 1).