Boolean analogues of the Pascal triangle with maximal possible number of ones

Abstract

Abstract The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array Ts formed by s(s+1)2 $\begin{array}{} \displaystyle \frac{s(s+1)}{2} \end{array}$ elements of the field GF(2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ s(s+1)3 $\begin{array}{} \displaystyle \frac{s(s+1)}{3} \end{array}$⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.