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 .
The number of relatively prime subsets of .
The number of squares and sets
The number of topologies on a finite set.
The -adic valuations of sequences counting alternating sign matrices.
The p-adic valuation of k-central binomial coefficients
The Pell Sequence of Order K, Multinomial Coefficients, and Probability
The Pfaffian transform.
The positivity problem for fourth order linear recurrence sequences is decidable
The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.