Abstract
We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.
Highlights
The first investigations on discrete lines are dated back to E.B
Morse (MH40) who introduced the terminology of sturmian sequences, for the ones defined on a two-letter alphabet and coding lines with irrational slope
Given two integer points of Z2 ( called pixels in the discrete geometry literature (CM91)), how many naive discrete segments link these points? In other words, given L ∈ N and h ∈ N, how much is s(L, h) = #{w ∈ {0, 1}L, |w|1 = h and w balanced}? We exhibit a recurrence relation on s(L, h) and generating functions and we study the asymptotic behaviour of the maps s
Summary
The first investigations on discrete lines are dated back to E.B. Christoffel (Chr75), A. Morse (MH40) who introduced the terminology of sturmian sequences, for the ones defined on a two-letter alphabet and coding lines with irrational slope. These works gave the first theoretical framework for discrete lines. In (dLdL05), the De Lucas investigated the number p(L) of balanced palindrome words of length L ∈ N, that is the balanced words coding a symmetrical discrete segments of length L. We focus on the number p(L, h) of balanced palindromes of given length and height for which we exhibit a recurrence relation and a generating function
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