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Greedy and lazy representations in negative base systems

Tomáš Hejda, Zuzana Masáková, Edita Pelantová (2013)

Kybernetika

We consider positional numeration systems with negative real base - β , where β > 1 , and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal ( - β ) -representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base β 2 with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...

Integers with a maximal number of Fibonacci representations

Petra Kocábová, Zuzana Masáková, Edita Pelantová (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the properties of the function R ( n ) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers F k . We determine the maximum and mean values of R ( n ) for F k ≤ n < F k + 1 .

Integers with a maximal number of Fibonacci representations

Petra Kocábová, Zuzana Masáková, Edita Pelantová (2010)

RAIRO - Theoretical Informatics and Applications

We study the properties of the function R(n) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers Fk. We determine the maximum and mean values of R(n) for Fk ≤ n < Fk+1.

New bounds on the length of finite pierce and Engel series

P. Erdös, J. O. Shallit (1991)

Journal de théorie des nombres de Bordeaux

Every real number x , 0 &lt; x ≤ 1 , has an essentially unique expansion as a Pierce series : x = 1 x 1 - 1 x 1 x 2 + 1 x 1 x 2 x 3 - ⋯ where the x i form a strictly increasing sequence of positive integers. The expansion terminates if and only if x is rational. Similarly, every positive real number y has a unique expansion as an Engel series : y = 1 y 1 - 1 y 1 y 2 + 1 y 1 y 2 y 3 + ⋯ where the y i form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates...

On multiplicatively dependent linear numeration systems, and periodic points

Christiane Frougny (2002)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

On multiplicatively dependent linear numeration systems, and periodic points

Christiane Frougny (2010)

RAIRO - Theoretical Informatics and Applications

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

Currently displaying 21 – 40 of 49