A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers

On the Harmonic and Hyperharmonic Fibonacci Numbers

In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is finite sums of reciprocals of Fibonacci numbers. We obtain spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers.

On the Norms of Circulant and $r-$Circulant Matrices With the Hyperharmonic Fibonacci Numbers

In this paper, we study norms of circulant and $r-$circulant matrices involving harmonic Fibonacci and hyperharmonic Fibonacci numbers. We obtain inequalities by using matrix norms.

与帕斯卡三角和斐波那契数列相关的三对角实对称矩阵

Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence

本研究推导了一类三对角实对称矩阵特征多项式的三项递推关系与闭式解，揭示其系数与帕斯卡三角对角元的关联，给出特征值包含关系的充分条件，并提出关联斐波那契数列的开放问题。

Symmetric Numerical Semigroups Generated by Fibonacci and Lucas Triples

The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) 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.

Convolution identities for Tribonacci numbers with symmetric formulae

Many kinds of convolution identities have been considered about several numbers, including Bernoulli, Euler, Genocchi, Cauchy, Stirling, and Fibonacci numbers. The well-known basic result about Bernoulli numbers is due to Euler. The convolution identities have been studied. In this paper, we give convolution identities for Tribonacci numbers without binomial coefficients and with binomial coefficients. Convolution identities of Fiboancci numbers or Lucas numbers can be expressed in the form of linear combinations of Fibonacci numbers and Lucas numbers only. Fibonacci numbers and Lucas numbers …

Chords of an ellipse, Lucas polynomials, and cubic equations

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of Price's ellipse proof, and a reorganized proof, with some new ideas, designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano's solution of the cubic equation and Newton's theorem on power sums, and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. …