scholarly journals Riordan matrices and sums of harmonic numbers

2011 ◽  
Vol 5 (2) ◽  
pp. 176-200 ◽  
Author(s):  
Emanuele Munarini
Keyword(s):  
Stirling Numbers ◽  
Catalan Numbers ◽  
Subject Classification ◽  
Harmonic Numbers ◽  
Lucas Numbers ◽  
Primary 05A15 ◽  
Secondary 05A10 ◽  
General Identity ◽  
2000 Mathematics Subject Classification ◽  
Fibonacci And Lucas Numbers

We obtain a general identity involving the row-sums of a Riordan matrix and the harmonic numbers. From this identity, we deduce several particular identities involving numbers of combinatorial interest, such as generalized Fibonacci and Lucas numbers, Catalan numbers, binomial and trinomial coefficients, Stirling numbers. 2000 Mathematics Subject Classification: Primary: 05A15; Secondary: 05A10.

Delving Deeper: The Ubiquitous Catalan Numbers

2006 ◽  
Vol 100 (3) ◽  
pp. 184-188
Author(s):  
Thomas Koshy
Keyword(s):  
Catalan Numbers ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers ◽  
Mathematical Activities

Just as Fibonacci and Lucas numbers are a great source of fun and excitement for both amateurs and professionals alike (Askey 2005, Koshy 2002), so are the less well-known Catalan numbers. They too are excellent candidates for mathematical activities such as experimentation, exploration, and conjecture. I was surprised to see that, like Fibonacci and Lucas numbers, Catalan numbers seemed to show up in several problems I had assigned to students over the years.

scholarly journals Polynomials related to harmonic numbers and evaluation of harmonic number series II

2011 ◽  
Vol 5 (2) ◽  
pp. 212-229 ◽  
Author(s):  
Ayhan Dil ◽  
Veli Kurt
Keyword(s):  
Subject Classification ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Number Series ◽  
Exponential Polynomials ◽  
2000 Mathematics Subject Classification

In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers. 2000 Mathematics Subject Classification. 11B73, 11B75, 11B83.

scholarly journals Fascinating Double Angle Formulas of the Mulatu Numbers

2016 ◽  
Vol 4 (1) ◽  
pp. 25-29
Author(s):  
Mulatu Lemma ◽  
Agegnehu Atena ◽  
Tilahun Muche
Keyword(s):  
Subject Classification ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

The Mulatu numbers were introduced in [1]. The numbers are sequences of numbers of the form: 4,1,5,6,11,17,28,45... The numbers have wonderful and amazing properties and patterns.In mathematical terms, the sequence ofthe Mulatu numbers is defined by the following recurrence relation:The double Angel Formulas for Fibonacci and Lucas numbers are given by the following formulasrespectively. Mathematical Subject Classification: 11

scholarly journals On New Formulas of Fibonacci and Lucas Numbers Involving Golden Ratio Associated with Atomic Structure in Chemistry

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1334
Author(s):  
Rifat Battaloglu ◽  
Yilmaz Simsek
Keyword(s):  
Atomic Structure ◽  
Generating Functions ◽  
Functional Equations ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.

scholarly journals Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2002
Author(s):  
Necdet Batir ◽  
Anthony Sofo
Keyword(s):  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Lucas Numbers ◽  
Finite Sums ◽  
Fibonacci And Lucas Numbers ◽  
Finite Sum

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

BASIS FOR SYNTHESIS OF SPIRAL LATTICE QUASICRYSTALS

1989 ◽  
Vol 03 (14) ◽  
pp. 1071-1085 ◽  
Author(s):  
L. A. BURSILL ◽  
GEORGE RYAN ◽  
XUDONG FAN ◽  
J. L. ROUSE ◽  
JULIN PENG ◽  
...  
Keyword(s):  
Experimental Error ◽  
Theoretical Models ◽  
Cell Number ◽  
Close Packing ◽  
Helianthus Tuberosus ◽  
Lucas Numbers ◽  
Average Growth ◽  
Left And Right ◽  
Seed Cell ◽  
Fibonacci And Lucas Numbers

Observations of the sunflower Helianthus tuberosus reveal the occurrence of both Fibonacci and Lucas numbers of visible spirals (parastichies). This species is multi-headed, allowing a quantitative study of the relative abundance of these two types of phyllotaxis. The florets follow a spiral arrangement. It is remarkable that the Lucas series occurred, almost invariably, in the first-flowering heads of individual plants. The occurrence of left-and right-handed chirality was found to be random, within experimental error, using an appropriate chirality convention. Quantitative crystallographic studies allow the average growth law to be derived (r = alτ−1; θ = 2πl/(τ + 1), where a is a constant, l is the seed cell number and τ is the golden mean [Formula: see text]). They also reveal departures from classical theoretical models of phyllotaxis, taking the form of persistent oscillations in both divergence angle and radius. The experimental results are discussed in terms of a new theoretical model for the close-packing of growing discs. Finally, a basis for synthesis of (inorganic) spiral lattice structures is proposed.

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

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