REGULARITIES OF NUMBERS IN THE FIBONACHI TRIANGLE CONSTRUCTED ON THE DEGREE TRANSFORMATIONS OF A SQUARE THREE MEMBERS

In this paper, it is shown that the Fibonacci triangle is formed from the elements of power transformations of a quadratic trinomial. It is binary structured by domains of rows of equal lengths, in which the sum of numbers forms a sequence of certain numbers. This sequence coincides with the transformed bisection of the classical sequence of Fibonacci numbers. The paper substantiates Pascal's rule for calculating elements in the lines of a Fibonacci triangle. The general relations of two forgings of numbers in lines of a triangle of Fibonacci for arbitrary values are received

MODELING OF ATOMIC SYSTEMS AND POSITIONING OF ELEMENTS OF NOBLE GASES OF THE PERIODIC TABLE BY PROPORTIONAL DIVISION METHOD

This paper studies regularities of proportional division, on the basis of which we show the possibility of effective application of the golden section method to modeling regularities of atomic systems and positioning of elements of noble gases of the periodic table. It is illustrated that by partial reconstruction of the Mendeleev tables, the elements of noble gases can be arranged along lines whose slope tangents in the coordinate system “the atomic number – the relative atomic mass” are in close agreement with the sequence of inverse Fibonacci numbers. It was shown that given the correct slope of axes, slope tangents of the corresponding lines does not change.

Repdigits as difference of two Fibonacci or Lucas numbers

In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)\in \{(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)\}.$ Further, if $L_{n}$ denotes the $n$-th Lucas number, then $L_{n}-L_{m}$ is a repdigit for $(n,m)\in\{(6,4),(7,4),(7,6),(8,2)\},$ where $n>m.$Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are $11,33,55,88$ and $555$; their representations are $11=F_{7}-F_{3},\33=F_{9}-F_{1}=F_{9}-F_{2},\55=F_{11}-F_{9}=F_{12}-F_{11},\88=F_{11}-F_{1}=F_{11}-F_{2},\555=F_{15}-F_{10}$ (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are $11,22$ and $44;$ their representations are $11=L_{6}-L_{4}=L_{7}-L_{6},\ 22=L_{7}-L_{4},\4=L_{8}-L_{2}$ (Theorem 3).

Fibonacci-Zeta infinite series associated with the polygamma functions

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.