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Abstract
Horadam introduced a generalized sequence of numbers, describing its key features and the special sub-sequences obtained from specific choices of initial parameters. This sequence and its sub-sequences are known as the Horadam, generalized Fibonacci, and generalized Lucas numbers, respectively. In the present study, we propose another new sequence, which satisfies a second-order recurrence relation. Further, we prove the Binet’s formula, some famous identities, and summation formulas for this new sequence. In particular, we demonstrate the interrelationships between our new sequence and the Horadam sequence.
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References
- Horadam A. F., A generalized Fibonacci sequence, Am. Math. Month., 68(5) (1961), 455-459.
- Horadam A. F., Generating functions for powers of a certain generalized sequence of numbers, Duke. Math. J., 32(3) (1965), 437-446.
- Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3(3) (1965), 161-176.
- Horadam, A. F., Special properties of the sequence Wn (a, b; p, q), Fibonacci Quart., 5(3)(1967), 424-434.
- Cerda-Morales G., On generalized Fibonacci and Lucas numbers by matrix methods, Hacet. J. Math. Stat., 42(2) (2013), 173-179.
- Zeilberger D., The method of creative telescoping, J. Symbolic Comput. , 11(3) (1991), 195-204
References
Horadam A. F., A generalized Fibonacci sequence, Am. Math. Month., 68(5) (1961), 455-459.
Horadam A. F., Generating functions for powers of a certain generalized sequence of numbers, Duke. Math. J., 32(3) (1965), 437-446.
Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3(3) (1965), 161-176.
Horadam, A. F., Special properties of the sequence Wn (a, b; p, q), Fibonacci Quart., 5(3)(1967), 424-434.
Cerda-Morales G., On generalized Fibonacci and Lucas numbers by matrix methods, Hacet. J. Math. Stat., 42(2) (2013), 173-179.
Zeilberger D., The method of creative telescoping, J. Symbolic Comput. , 11(3) (1991), 195-204