Linear numeration systems of order two

Conversion between Two Multiplicatively Dependent Linear Numeration Systems

Numeration Systems, Linear Recurrences, and Regular Sets

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2, . . . expresses a non-negative integer n as a sum n = ∑i j=0 ajuj . In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1, . . .) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the ...

Numeration Systems

On Properties of Representations in Certain Linear Numeration Systems

Given a ≥ b, let G0 = 1, G1 = a+ 1, and Gn+2 = aGn+1 + bGn for n ≥ 0. For each choice of a and b, we have a linear recurrence that defines a numeration system. Every positive integer n may be written as the sum of the Gn, with alphabet A = {0, 1, . . . a}, in one or more different ways. Let R(a,b)(n) be the function that counts the number of distinct representations of an integer as a sum of th...

Combinatorial and Arithmetical Properties of Linear Numeration Systems

with digits δl{0, 1} for 0 ≤ l ≤ L, where the digits are computed by the greedy algorithm: there is a unique integer L such that GL ≤ n < GL+1. Then n can be written as n = δLGL + nL with 0 ≤ nL < GL and by iterating this procedure with nL the expansion (1.3) is obtained. An extensive description of digital expansions with respect to linear recurring base sequences is given in [15, 19, 20, 21]....