The joint distribution of the binary digits of integer multiples
The Josephus problem
We give explicit non-recursive formulas to compute the Josephus-numbers and and explicit upper and lower bounds for (where ) which differ by (for the bounds are even better). Furthermore we present a new fast algorithm to calculate which is based upon the mentioned bounds.
The jumping champions of the Farey series
The -binomial transforms and the Hankel transform.
The Kazandzidis supercongruences. A simple proof and an application
The k-Fibonacci matrix and the Pascal matrix
We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.
The Lagrange inversion formula and divisibility properties.
The largest subset in [1,n] whose integers have pairwise l.c.m. not exceeding n, II
The last digit of and .
The lonely runner with seven runners.
The Lucas congruence for Stirling numbers of the second kind
0. Introduction. The numbers introduced by Stirling in 1730 in his Methodus differentialis [11], subsequently called “Stirling numbers” of the first and second kind, are of the greatest utility in the calculus of finite differences, in number theory, in the summation of series, in the theory of algorithms, in the calculation of the Bernstein polynomials [9]. In this study, we demonstrate some properties of Stirling numbers of the second kind similar to those satisfied by binomial coefficients; in...
The minimal density of a letter in an infinite ternary square-free word is 883/3215.
The minimal density of a letter in an infinite ternary square-free word is .
The minimal range of additive h-bases.
The monotone Poisson process
The coefficients of the moments of the monotone Poisson law are shown to be a type of Stirling number of the first kind; certain combinatorial identities relating to these numbers are proved and a new derivation of the Cauchy transform of this law is given. An investigation is begun into the classical Azéma-type martingale which corresponds to the compensated monotone Poisson process; it is shown to have the chaotic-representation property and its sample paths are described.
The multiplicity of binary recurrences
The number of (2,3)-sum-free subsets of {1,...,n}
The number of binomial coefficients in residue classes modulo p and .
The number of k-sums of abelian groups of order k
The number of relatively prime subsets and phi functions for .