The k-nacci triangle and applications

Abstract: A generalization of the classical Fibonacci numbers Fn is the k-generalized Fibonacci numbers F n for n ≥ 2 − k whose first k terms are 0, ... , 0, 1 and each term afterward is the sum of the preceding k terms. In this article, we first introduce the k-nacci triangle to derive an explicit formula of the nth k-generalized Fibonacci number. Second, we also introduce the k-generalized Pascal triangle for deriving the formula of the k-generalized Fibonacci numbers.

Such a sequence is also called k-step Fibonacci sequence or the Fibonacci k-sequence. Clearly for k = 2, we obtain the well-known Fibonacci numbers F (2) n = F n , for k = 3, the tribonacci numbers F (3) n = T n , for k = 4, the tetranacci numbers F (4) n =  n , and for k = 5, the pentanacci numbers F (5) n =  n .
In general case, the first k + 1 non-zero terms in F (k) n are powers of two, namely while the next term in the above sequence is F (k) k+2 = 2 k − 1.

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Fibonacci numbers and their generalizations have many interesting properties and applications to almost every fields of science and art. Pascals triangle has been explored for links to the Fibonacci sequence as well as to generalized sequences. In this paper, the authors give some connections of the coefficients in the multinomial expansion with the generalized Fibonacci numbers. They also construct the k-generalized Pascal triangle to derive the formula of the nth k-generalized Fibonacci numbers.
The Fibonacci numbers and their generalizations have many interesting properties and applications to almost every fields of science and art (e.g. see Debnart, 2011;Koshy, 2001;Vajda, 1989).
It is well-known that the Fibonacci number F n can be derived by the summing of elements on the rising diagonal lines in Pascal's triangle where ⌊x⌋ is the largest integer not exceeding x, see Koshy (2001, chapter 12). Wong and Maddocks (1975) generalized the Pascal's triangle and showed that sums of elements on the rising diagonal lines in their triangle give the tribonacci number T n (some authors called this triangle that the tribonacci triangle, e.g. see Alladi and Hoggatt (1977), Kuhapatanakul (2012)).
There is yet another triangular array that yields the various tribonacci numbers. Feinberg (1964) used the trinomial expansions of (1 + x + x 2 ) n for n ≥ 0 and showed that the rising diagonal sums of this trinomial coefficient array also yield the tribonacci numbers. Kuhapatanakul and Sukruan (2014) have shown the n-triboacci triangle similar to Pascal's triangle and derived an explicit formula for the tribonacci numbers. The n-tribonacci triangle is an array of the shape They showed that the sums of all elements in the n-tribonacci triangle give the nth tribonacci number. The expansion for the tribonacci number T n in terms of binomial coefficients, see Kuhapatanakul (2012), Kuhapatanakul and Sukruan (2014), Shannon (1977), as the following Philippou and Muwafi showed in (1982) that the F (k) n can be written in the form The following "Binet-like" formula for F (k) n appears in Dresden and Du (2014): In this article, we extend the result of Kuhapatanakul and Sukruan (2014) on the n-triboacci triangle to the k-nacci triangle for n, and derive the nth k-generalized Fibonacci numbers. We also construct the k-generalized Pascal's triangle to derive the nth k-generalized Fibonacci numbers. (1.1)

The k-nacci triangle
Define the symbol C q i, j as the coefficient of x j in the multinomial expansion of (1 + x + x 2 + ⋯ + x q ) i for q ≥ 1 and i, j ≥ 0, i.e.
Using the classical binomial coefficient, we get that or We give the arrays of the coefficients of x j in the multinomial expansion for q = 2, 3, 4 to show in the Figures 1-3, respectively, see also Sloane (2011) as A027907, A008287, A035343.
(2.1)   Some well-known properties of the multinomial coefficient arrays: (1) Every row is symmetric about a vertical line through the middle, i.e.
(2) Any interior number in each row, the exception of the first two rows, can be obtained from the preceding row, i.e.
Next, we introduce the k-nacci triangle for a positive integer n.
Definition 1 Let n and k ≥ 3 be positive integers. The k-nacci triangle for n is an array that each element in row i and column j is products of C k−2 i, j and For clarity, we give the examples of the k-nacci triangle for k = 3, 4, 5.
It is interesting that sums of all elements in the k-nacci triangle for n give the nth k-nacci number. We will give some examples.

Example 1
(1) In the 4-nacci triangle for n (a) Substituting n = 10, we get and then sums of all elements is equal to 208, which is the 10th tetranacci number  10 . and sums of all elements is equal to 5600, which is the 15th tetranacci number  15 .

Example 2
(2) In the 5-nacci triangle for n (a) Substituting n = 10, we get and sums of all elements is equal to 236, which is the 10th pentanacci number  10 .
(b) Substituting n = 15, we get and sums of all elements is equal to 6930, which is the 15th pentanacci number  15 .
We will in fact prove that the sums of all elements in the k-nacci triangle for n give the nth k-nacci number.
Let k ≥ 3. Denote S n (i) as the sums of elements in the ith row of k-nacci triangle for a positive integer n, that is, The S n (i) can write in the recurrent relation.

Lemma 1 Let n, i be positive integers. Then
Proof Set C i, j : = C k−2 i, j and we see that C i, j = 0 when j > (k − 1)(i − 1). We have as desired. ✷ Now, we state an explicit formula for F (k) n by summing all row sums of the k-nacci triangle for n and prove the following theorem.
Theorem 1 Let k ≥ 3 and S n (i) as defined in this section. Then Proof We will prove by induction on n. We see that (2.6) Assume that (2.6) holds for all integers n = 0, 1, 2, … , k − 1. By the inductive hypothesis and Lemma 1, we get Thus, the proof is complete. ✷ We can rewrite Equation (2.6) in terms of binomial coefficients using Equation (2.4).

The k-generalized Pascal's triangle
Throughout the section, the integer k ≥ 2 will be fixed.
Definition 2 Let n, i be integers with n ≥ 0. Define where C k (n, 0) = C k (n, n) = 1, and C k (n, i) = 0 for i < 0 or i > n.
Definition 3 Denote the k-generalized Pascal's triangle as follows: Well-known examples of C k (n, i) for k = 2, 3 are wrote in terms of binomial coefficients (3.1) It is to see that the 2-generalized Pascal's triangle is the classical Pascal's triangle and the 3-generalized Pascal's triangle is the generalized Pascal's triangle which is defined by Wong and Maddocks (1975).
For convenience, we arrange the elements of the k-generalized Pascal's triangle to form a leftjustified triangular array as follows: The k-generalized Pascal's triangle For clarity, we also give the following examples of the k-generalized Pascal's triangles for k = 4, 5.  We conjecture that the sums of elements on each rising diagonal line in the k-generalized Pascal's triangle gives the k-generalized Fibonacci number.
Theorem 2 Let k ≥ 2 be fixed, and let n be non-negative integer. Then Proof For k = 2, C 2 (n, i) is a binomial coefficient, the Equation (3.2) becomes the Equation (1.1). Suppose k ≥ 3. We will prove this result by induction on n. It is easy to see that, for 1 ≤ m ≤ k − 1, Now, we assume Equation (3.2) holds for n > 1 and prove that it holds for n + 1. Using the definition of F (k) n and the inductive hypothesis, we get that Thus, the proof of result is complete. ✷ Next, we give an alternative definition of C k (n, i) in terms of binomial coefficients. We begin provide the following lemma which will be used in the proof.
Lemma 2 Let n be non-negative integer. Then Proof Using the Pascal's identity, we get the parts (i) and (