ON THE COMPONENTS OF SOME SPECIAL SEQUENCES
Dişkaya, Orhan | Menken, Hamza
The Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences {Fn}, {Ln}, {Pn}, {P Ln}, {Jn} and {JLn} are defined by two order recurrences for n ≥ 0, respectively, Fn+2 = Fn+1 + Fn, Ln+2 = Ln+1 + Ln, Pn+2 = 2Pn+1 + Pn, P Ln+2 = 2P Ln+1 + P Ln, Jn+2 = Jn+1 + 2Jn, JLn+2 = JLn+1 + 2JLn, with the initial conditions, respectively, F0 = 0, and F1 = 1, L0 = 2, and L1 = 1, P0 = 0, and P1 = 1, P L0 = 2, and P L1 = 1, J0 = 0, and J1 = 1, JL0 = 2, and JL1 = 1. In this work we define new component sequences which generalize the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences with different initial conditions. We give some identities of these component sequences. Also, the Binet-like formulas, the generating functions and the exponentia...
SOME IDENTITIES FOR SEQUENCES OF BINOMIAL SUMS OF GENERALIZED PADOVAN IDENTITIES INCLUDING POWERS AND BINOMIAL COEFFICIENTS
Dişkaya, Orhan
In this work, we investigate several identities of the generalized Padovan sequence, including binomial coefficients, using the method of ordinary power series generat- ing functions. We explore the characteristics and identities of generalized Padovan sequences, examining binomial coefficients and other extended identities through this method. Nevertheless, specific cases of third-order sequences have been ex- plored and applied in this context. By employing various techniques, we aim to derive new results concerning third-order recurrence relations, enhancing our com- prehension of these complex sequences.
A NEW GENERALIZATION OF THE PADOVAN SEQUENCE
Dişkaya, Orhan | Menken, Hamza
Integers number sequences in mathematics are one of the subjects with the most ap- plication area. The Fibonacci number sequence has applications in many branches of science such as nature, anatomy, botany, zoology, art, music, analysis, physics, astronomy, chemistry, biology and computers. Since the positive real root of the Fibonacci number sequence gives the golden ratio, it has many applications. Many scientists deal with Fibonacci sequence and its generalizations in recent years. Some of these generalizations are number sequences such as Lucas, Pell, and Jacobthal [4, 5, 6]. In this study, the Padovan numbers sequence, which has a third-order characteristic equation, and some of its generalizations are examined. The Padovan sequence {Pn} is defined by the third order recurren...
COMPOSITIONS OF POSITIVE INTEGERS AND THE PADOVAN NUMBERS
Dişkaya, Orhan | Menken, Hamza
The Padovan sequence {Pn}n≥0 is defined by the third order recurrence (1) Pn+3 = Pn+1 + Pn with the initial conditions P0 = 1, P1 = 0 and P2 = 1. The Padovan sequence appears as sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [1]. For relevance, we consider as P−2 = P−1 = 0. In [2] the Padovan polynomial sequence {Pn(x)}n≥0 is defined by a third order recurrence (2) Pn+3(x) = xPn+1(x) + Pn(x) with the initial conditions P0(x) = 1, P1(x) = 0 and P2(x) = x. For relevance, we consider as P−2(x) = P−1(x) = 0. To simplify notation, take Pn(x) = Pn. A composition of an integer n is a representation of n as a sum of positive integers, for example the eight compositions of 4 are as follows: 4, 3+1, 1+3, 2+2, 2+1+1, 1 + 2 + 1, 1 + 1 + 2, 1 + 1 + 1 + 1. A partition ...
SOME NOTES ON THE PLASTIC CONSTANT
Dişkaya, Orhan | Menken, Hamza
The Padovan sequence is named after Richard Padovan who attributed its dis- covery to Dutch architect Hans van der Laan in his 1994 essay Dom. The Padovan sequence {Pn}n≥0 is defined by (1) P0 = P1 = P2 = 1 and Pn+3 = Pn+1 + Pn for all n ≥ 0. Here, Pn is the nth Padovan number. First few terms of this sequence are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21. The ratio of successive the Padovan number converges to the plastic constant. As n gets larger, it appers that PPn+1 n approaches a limit, namely, 1, 32471795724474602596 . . . The plastic number p (also known as the plastic constant, the plastic ratio, the platin number and the minimal Pisot number) is a mathematical constant which is the unique real solution of the cubic equation x3 − x − 1 = 0. It has the exact value...
ON THE SEQUENCE OF GELL NUMBERS
Dişkaya, Orhan | Menken, Hamza
The Pell sequence {Pn}n≥0 is defined by the initial values P0 = 0 and P1 = 1 and the recurrence relation Pn+2 = 2Pn+1 + Pn, n ≥ 0. In this work we introduce a new class of Gell numbers which generalizes Pell num- bers. We derive Binet-like formula, generating function and some identities of Gell numbers.
ON A GENERALIZATION OF THE HOSOYA TRIANGLE
Menken, Hamza | Dişkaya, Orhan
In this talk, we introduce the Fibonacci polynomial triangle, inspired by the struc- ture of the Hosoya triangle, utilizing Fibonacci polynomials. Additionally, we present and prove several identities and properties associated with the Fibonacci polynomial triangle.