A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most $1$. In this paper, the existence of balanced binary triangles of size $n$, for all positive integers $n$, is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.

Source : oai:HAL:hal-01463665v3

Volume: Vol 19 no. 3

Section: Combinatorics

Published on: November 28, 2017

Submitted on: February 14, 2017

Keywords: periodic orbits, generalized Pascal triangles, periodic triangles, Steinhaus Problem, balanced triangles,binary triangles, Steinhaus triangles,MSC2010: 05B30, 11B75, 05A05, 11A99, 05A99,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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