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...
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.