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The Josephus problem

Lorenz Halbeisen, Norbert Hungerbühler (1997)

Journal de théorie des nombres de Bordeaux

We give explicit non-recursive formulas to compute the Josephus-numbers j ( n , 2 , i ) and j ( n , 3 , i ) and explicit upper and lower bounds for j ( n , k , i ) (where k 4 ) which differ by 2 k - 2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j ( n , k , i ) which is based upon the mentioned bounds.

The k-Fibonacci matrix and the Pascal matrix

Sergio Falcon (2011)

Open Mathematics

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 Lucas congruence for Stirling numbers of the second kind

Roberto Sánchez-Peregrino (2000)

Acta Arithmetica

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 monotone Poisson process

Alexander C. R. Belton (2006)

Banach Center Publications

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.

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