Generalized (k, r)–Fibonacci Numbers
نویسنده
چکیده
In this paper, and from the definition of a distance between numbers by a recurrence relation, new kinds of k–Fibonacci numbers are obtained. But these sequences differ among themselves not only by the value of the natural number k but also according to the value of a new parameter r involved in the definition of this distance. Finally, various properties of these numbers are studied.
منابع مشابه
ON THE GENERALIZED ORDER-k FIBONACCI AND LUCAS NUMBERS
In this paper we consider the generalized order-k Fibonacci and Lucas numbers. We give the generalized Binet formula, combinatorial representation and some relations involving the generalized order-k Fibonacci and Lucas numbers.
متن کاملA Combinatoric Proof and Generalization of Ferguson’s Formula for k-generalized Fibonacci Numbers
Various generalizations of the Fibonacci numbers have been proposed, studied and applied over the years (see [5] for a brief list). Probably the best known are the k-generalized Fibonacci numbers F (k) n (also known as the k-fold Fibonacci, k-th order Fibonacci, k-Fibonacci or polynacci numbers), satisfying F (k) n = F (k) n−1 + F (k) n−2 + . . .+ F (k) n−k , n ≥ k , F (k) n = 0 , 0 ≤ n ≤ k − 2...
متن کاملOn Families Of Bipartite Graphs Associated With Sums Of Generalized Order-k Fibonacci And Lucas Numbers
In this paper, we consider the relationships between the sums of the generalized order-k Fibonacci and Lucas numbers and 1-factors of bipartite graphs. 1. Introduction We consider the generalized order k Fibonacci and Lucas numbers. In [1], Er de ned k sequences of the generalized order k Fibonacci numbers as shown: g n = k X j=1 g n j ; for n > 0 and 1 i k; (1.1) with boundary conditions for 1...
متن کاملEnergy of Graphs, Matroids and Fibonacci Numbers
The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.
متن کاملRestricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci Numbers
In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 132, 213, and 123 is equal to the Fibonacci number Fn+1. We use generating function and bijective techniques to give other sets of pattern-avoiding permutations which can be enumerated in terms of Fibonacci or k-generalized Fibonacci numbers.
متن کاملUsing Generalized Fibonacci Sequences for Solving the One-Dimensional LQR Problem and its Discrete-Time Riccati Equation
In this article we develop a method of solving general one-dimensional Linear Quadratic Regulator (LQR) problems in optimal control theory, using a generalized form of Fibonacci numbers. We find the solution R (k) of the corresponding discrete-time Riccati equation in terms of ratios of generalized Fibonacci numbers. An explicit Binet type formula for R (k) is also found, removing the need for ...
متن کامل