Catalan transform of the k-Fibonacci sequence

Catalan transform of the k-Fibonacci sequence

´ Sergio Falcon a,∗ and Angel Plaza a a Department

of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), Spain

Abstract In this paper we apply the Catalan Transform to the k-Fibonacci sequence finding different integer sequences, some of wich are indexed in OEIS and others not. After we apply the Hankel transform to the Catalan transform of the k-Fibonacci sequence and obtain an unusual property. Key words: k-Fibonacci numbers, Catalan Numbers, Catalan Transform, Hankel Transform.

1

Introduction

Classical Fibonacci numbers have been very used in as different Sciences as the Biology, Demography or Economy [7]. Recently they have been applied even ∗ Corresponding author. Address: University of Las Palmas de Gran Canaria, Department of Mathematics, 35017-Las Palmas de Gran Canaria, Spain. Tel.: +34 928 45 88 27; fax: +34 928 45 88 11. Email address: [email protected] (Sergio Falcon).

Preprint submitted to

8 January 2013

in the high-energy physics [10–13]. But there exist generalizations of these numbers given by researchs as Horadam [8] and recently by we oursef [3–5]. Now we present this last generalization, so called the k-Fibonacci numbers.

1.1 k-Fibonacci numbers

For any positive real number k, the k-Fibonacci sequence, say {Fk,n }n∈N , is defined recurrently by Fk,n+1 = k Fk,n + Fk,n−1 for n ≥ 1

(1)

with initial conditions Fk,0 = 0 and Fk,1 = 1 For k = 1, classical Fibonacci sequence is obtained and for k = 2, Pell sequence appears. The well-known Binet’s formula in the Fibonacci numbers theory [3,8,16] allows us to express the k-Fibonacci numbers in function of the roots r1 and r2 of the characteristic equation, associated to the recurrence relation r2 = k r + 1:

Fk,n =

r1n − r2n r1 − r2

(2)

If σ denotes the positive root of the characteristic equation, σ = the general term may be written in the form Fk,n =

k+

√

k2 + 4 , 2

σ n − (−σ)−n , and it is σ + σ −1

verified the limit of the quotient of two terms is:

lim

n→∞

Fk,n+r = σr Fk,n

(3)

2

In particular, if k = 1, then σ is the Golden Ratio, φ =

√ 1+ 5 . 2

In addition, the

general term of the k-Fibonacci sequence may be obtained by the formula:

1

Fk,n =

2n−1

n−1 ⌊X 2 ⌋

i=0

!

n k n−2i−1 (k 2 + 4)i 2i + 1

(4)

or, equivalently, by:

Fk,n =

n−1 ⌊X 2 ⌋

i=0

!

n − 1 − i n−1−2i k i

(5)

1.2 Catalan numbers

Catalan numbers are described by [1] 1 2n Cn = n+1 n

!

(6)

and the ordinary generating function of the correponding Catalan sequence is given by c(x) =

1−

√

1 − 4x 2x

Formula (6) can be written as Cn =

(7)

(2n)! (n + 1)!n!

Finally, a recurrence relation for C(n) is obtained from

Cn+1 2(2n + 1) = [17] Cn n+2

The first few Catalan numbers, for n = 0, 1, 2, . . ., are {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...} cited in [15], from now on OEIS, as A000108. 3

1.3 Catalan transform

We define the Catalan transform of the sequence {an } as the sequence {c[an ]} such that n X

!

j 2n − j c[an ] = aj n−j j=1 2n − j The matrix associated to this transformation is the lower-triangular matrix 

     C=     

1 1 2 5 14 42 .. .

1 2 5 14 42 .. .

1 3 9 28 .. .

 1 4 14 .. .

1 5 .. .

1 .. .

..

.

          

The first column is the sequence of the Catalan numbers, the second is the selfconvolution of the first and the following column sequences are the convolution of the preceding column with the first.

In [1] is proved that the terms ci,j of this matrix are ci,j =

 j   X r/2 j (−1)i+r 22i−j i r r=0

1.4 Binomial Transform

An example of tranformations that operate on integer sequences is the so-called binomial transform [19], which associates to the sequence with with general term an the sequence with general term bn where

bn =

n   X n j=0

j

aj

The matrix of the coefficients of this transformation is the well-known matrix

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

     B=     

1 1 1 1 1 1 .. .

1 2 3 4 5 .. .

1 3 6 10 .. .

 1 4 10 .. .

1 5 .. .

1 .. .

..

.

      (Pascal’s triangle).     

From the second column, each column sequence is the convolution of the preceding column with the constant sequence {1, 1, 1, 1, . . .} n   X n This transformation is invertible with an = (−1)n−j bj j j=0

In particular, the Binomial transform of the Catalan numbers is {B[cn ]} = {1, 2, 5, 15, 51, 188, 731, 2950, . . .}, A007137 in OEIS. The terms of this sequence verify the recurrence relation (n + 2)an+2 = (6n + 4)an+1 − 5n an

1.5 Binomial transform of the Catalan transform

We can applied the Binomial transform to sequence generated by mean of the Catalan transform of an integer sequence {an } so that BC[an ] =

j n X X j=1 i=1

   n 2j − i i an j−i 2j − i j

Then

• If {an } is the constant sequence {an } = {1}, then {BC[1]} = {1, 4, 14, 50, 187, 730, 2949, 12234, 51821, 223190, . . .}. This sequence is the self-convolution of the sequence generated from the Binomial transform of the Catalan numbers {B[cn ]}. Moreover, each term of this sequence is equal to the correspondent term of the sequence {B[cn ]} less 1. Consequently, its terms verify the relation (n + 2)bn+2 = (6n + 4)bn+1 − 5n bn + 2 • If {an } is the sequence of the powers of 2, {2n }, then

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{BC[2n−1 ]} = {1, 5, 22, 97, 436, 1994, 9241, 43257, 204052, 968440, 4619011, . . .}, A129164 in OEIS • If {an } is the sequence of the natural numbers, then {BC[n]} = {1, 5, 21, 86, 355, 1488, 6335, 27352, 119547, 528045, 2353791, . . .}, A039919 in OEIS

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The Catalan transform of the k-Fibonacci sequence

Following [1], we define the Catalan Transform of the k-Fibonacci sequence {Fk,n } as CFk,n =

n X j=0

  2n − j j Fk,j for n ≥ 1 2n − j n − j

(8)

with CFk,0 = 0 The first members of these sequence, that is to say, Catalan transform of the first k-Fibonacci numbers, are the polynomials in k:

• CFk,1 =

1 X

  j 2−j Fk,j = 1 2−j 1−j

• CFk,2 =

2 X

  4−j j Fk,j = k + 1 4−j 2−j

• CFk,3 =

3 X

  j 6−j Fk,j = k 2 + 2k + 3 6−j 3−j

1

1

1

• CFk,4 = k 3 + 3k 2 + 7k + 8 • CFk,5 = k 4 + 4k 3 + 12k 2 + 22k + 24 • CFk,6 = k 5 + 5k 4 + 18k 3 + 43k 2 + 73k + 75

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We can write equation (8) as the product of the lower triangular matrix C and the n × 1 matrix Fk as we indicate next:    1 CFk,1  CF   1 1   k,2     2  CFk,3   2     CFk,4  =  5 5     CFk,5   14 14     CFk,6   42 42    .. .. .. . . .

1 3 9 28 .. .

 Fk,1  F    k,2       Fk,3      ·  Fk,4       Fk,5       Fk,6     .. .. . .  

1 4 14 .. .

1 5 .. .

1 .. .

The entries of the matrix C verify the recurrence relation Ci,j =

i−1 X

Ci−1,r and

r=j−1

the first column equales the second for i > 1 and they are the Catalan numbers. Lower triangular matrix Cn,n−j is known as Catalan triangle and its elements verify (2n − j)!(j + 1) the formula Cn,n−j = with 0 ≤ j ≤ n. (n − j)!(n + 1)! n−j

c′n,j

This tranformation is invertible with  1  -1 1   0 -2 1   1 -3 1  is   0 3 -4 1   -1 6 -5 1   0 -4 10 -6 ... ... ...

= (−1)





j n−j



and the inverse matrix

           

With the coefficients of the Catalan transform of the k-Fibonacci sequence we will form the following triangle: Table 1 Catalan triangle of the k-Fibonacci sequence CFk,1 CFk,2 CFk,3 CFk,4 CFk,5 CFk,6 CFk,7 ...

1 1 1 1 1 1 1 ...

1 2 3 4 5 6 ...

3 7 12 18 25 ...

7

8 22 43 72 ...

24 73 156 ...

75 246 ...

243 ...

First diagonal sequence {1, 1, 3, 8, 24, 75, 243, . . .} : A000958, is the Catalan Transform of the Fine sequence {1, 0, 1, 2, 6, 18, 57, 186, . . .} : A000957 (see [1]). Second diagonal sequence, A114495, is the self-convolution of the first sequence, A000958. From this point on, each diagonal sequence is the convolution of its preceding diagonal sequence and the first one. Finally, for k = 1, 2, 3, . . . we obtain the following Catalan transform of the kFibonacci sequence, Fk = {Fk,n , n ∈ N }: • CF1 = {0, 1, 2, 6, 19, 63, 215, 749, 2650, 9490, . . .}, indexed in OEIS as A109262, is the Catalan transform of the classical Fibonacci numbers. • CF2 = {0, 1, 3, 11, 42, 164, 649, 2591, 10408, 41998, . . .}, A143464 in OEIS is the Catalan transform of the Pell numbers. • CF3 = {0, 1, 4, 18, 83, 387, 1815, 8541, 40276, 190182, . . .}, A143646 in OEIS, is the Catalan transform of the 3–Fibonacci sequence {0, 1, 3, 10, 33, 109, . . .}. From this point, no other sequence is cited in OEIS. • CF4 = {0, 1, 5, 27, 148, 816, 4511, 24971, 138328, . . .} • CF5 = {0, 1, 6, 38, 243, 1559, 10015, 64373, 413878, . . .}

2.1 Generating function

x is the generating function of the k-Fibonacci poly1 − k x − x2 nomials [3] whereas the generating function of the Catalan numbers is c(x) = √ 1 − 1 − 4x . 2x Function fk (x) =

In [1] it is proved that if c(x) is the generating function of the sequence of the Catalan numbers {Cn } and A(x) is the generating function of the sequence {an }, then A(x ∗ c(x)) is the generating function of the Catalan Transform of this last

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sequence. Consequently, the generating function of the Catalan transform of the k-Fibonacci sequence {Fk,n } is √ 1 − 1 − 4x √ cfk (x) = fk (x ∗ c(x)) = 1 − k + (k + 1) 1 − 4x + 2x

3

(9)

Hankel transform

Let A = {a0 , a1 , a2 , . . .} be a sequence of real numbers [2,9]. Hankel Transform of the sequence A is the sequence of determinants Hn = Det[ai+j−2 ], i.e.,

a0 a1 Hn = a2 a 3 . . .

a1 a2 a3 a2 a3 a4 a3 a4 a5 a4 a5 a6 .. .. .. . . .

· · · · · · · · · · · · .. .

The Hankel determinant of order n of the sequence A is the upper-left n × n subdeterminant of Hn . The Hankel transform of the Catalan sequence is the sequence {1, 1, 1, . . .} [20] and the Hankel Transform of the sum of consecutive generalized Catalan numbers is the bisection of Classical Fibonacci sequence [14]. On the other hand, the element of order n ≥ 3 of the Hankel Transform of the k-Fibonacci sequence es null because each row of the determinant is a linear combination of the two preceding rows (1), according the definition of k-Fibonacci numbers. By this reason, the Hankel transform of the k-Fibonacci sequence {Fk,n } lacks interest. However, it is very interesting the study of the Catalan Transform of this sequence, as we will see in the sequel.

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Taking into account the Catalan transform of the k-Fibonacci sequence of the preceding subsection, we find out: HCF1 = Det[1] = 1 1 k+1 =2 HCF2 = k + 1 k 2 + 2k + 3 2 1 k+1 k + 2k + 3 2 3 2 HCF3 = =5 k+1 k + 2k + 3 k + 3k + 7k + 8 2 3 2 4 3 2 k + 2k + 3 k + 3k + 7k + 8 k + 4k + 12k + 22k + 24 We can continue in this form and then we will find that the Hankel transform of the Catalan transform of the k-Fibonacci sequence {Fk,n } is the sequence {1, 2, 5, . . .}, A001519 in OEIS

There is a wonderfull property that we will study next.

Main Theorem 1 Hankel transform of Catalan transform of the k-Fibonacci sequence is the bisection of classical Fibonacci sequence {1, 2, 5, 13, 34, 89, . . .} (A001519), independently of the value of k ∈ N . That is, {HCFk,n } = {F2n+1 }

Proof. Because the determinant Hn 6= 0, its matrix is nonsingular and it may factored as the product Ln · Un with Ln is a lower triangular matrix which main diagonal is {1, 1, 1, . . .} and the first column {CF1 , CF2 , CF3 , . . .} and Un is an upper triangular matrix which the first row is {CF1 , CF2 , CF3 , . . .} and the main diagonal 34 {1, 21 , 25 , 13 5 , 13 , . . .}. Then, as Hn = Det(Ln ) · Det(Un ) = Det(Un ) = a11 · a22 · · · ann ,

the sequence of determinants is A001519.

Finally, to make notice that this sequence is the complemmentary sequence of A001906 in the Fibonacci sequence, obtained as the Hankel Transform of the sequence of sums of two adjacent Catalan numbers A005807 [2].

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References

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