MathDB

Consider a triangular table of positive integers \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} The first row consists of odd numbers only. For

i>1,j\ge1

we have

a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.

If we get out of range with the second index, we consider such

a

to be zero (eg.

a_{22}=0+a_{11}+a_{12}

and

a_{37}=a_{25}+0+0

). Show that for every

i>1

there is

j\in\{1,\ldots,2i+1\}

such that

a_{ij}

is even.

Problem Statement