We generalize both the Fibonacci and Lucas numbers to the context of graph colorings, and prove some identities involving these numbers. As a corollary we obtain new proofs of some known identities involving Fibonacci numbers such as Fr+s+t = Fr+1Fs+1Ft+1 + FrFsFt − Fr−1Fs−1Ft−1.

The symmetric numerical semigroups S (Fa, Fb, Fc) and S (Lk, Lm, Ln) generated by three Fibonacci (Fa, Fb, Fc) and Lucas (Lk, Lm, Ln) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.

The purpose of the present paper is to prove that there are finitely many binomial coefficients of the form (f in certain binary recurrences, and give a simple method for the determination of these coefficients. We illustrate the method by the Fibonacci, the Lucas, and the Pell sequences. First, we transform both of the title equations into two elliptic equations and apply a theorem of Mordell ...

In this paper, we introduce a tiling approach to (p,q)-Fibonacci and (p,q)-Lucas numbers that generalize of the well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal ve Jacobsthal-Lucas numbers. We show nth number is interpreted as ways tile 1×n board with cells labeled 1,2,...,n using colored 1×1 squares 1×2 dominoes, where there are p kind colors for q dominoes. Then circular also present...

We generalize both the Fibonacci and Lucas numbers to the context of graph colorings, and prove some identities involving these numbers. As a corollary we obtain new proofs of some known identities involving Fibonacci numbers such as Fr+s+t = Fr+1Fs+1Ft+1 + FrFsFt − Fr−1Fs−1Ft−1.

We present a fast algorithm for computing large Fibonacci numbers. It is known that the product of Lucas numbers algorithm uses the fewest bit operations to compute the Fibonacci number Fn. We show that the number of bit operations in the conventional product of Lucas numbers algorithm can be reduced by replacing multiplication with the square operation.  2000 Elsevier Science B.V. All rights ...

We generalize both the Fibonacci and Lucas numbers to the context of graphcolorings, and prove some identities involving these numbers. As a corollary we obtain newproofs of some known identities involving Fibonacci numbers such asFr+s+t = Fr+1Fs+1Ft+1 + FrFsFt − Fr−1Fs−1Ft−1.

We generalize both the Fibonacci and Lucas numbers to the context of graph colorings, and prove some identities involving these numbers. As a corollary we obtain new proofs of some known identities involving Fibonacci numbers such as Fr+s+t = Fr+1Fs+1Ft+1 + FrFsFt − Fr−1Fs−1Ft−1.

In this paper we obtain formulas for certain sums of products involving multinomial coefficients and Fibonacci numbers. The sums studied here may be regarded as generalizations of the binomial transform of the sequence comprising the even-numbered terms of the Fibonacci sequence. The general formulas, involving both Fibonacci and Lucas numbers, give rise to infinite sequences that are parameter...

In this paper, we define and investigate a new class of bi-Bazilevic functions related to shell-like curves connected with Fibonacci numbers.  Furthermore, we find estimates of first two coefficients of functions belonging to this class. Also, we give the Fekete-Szegoinequality for this function class.