Question 195
An equilateral triangle of integer side length $n \ge 1$ is subdivided into small triangles of unit side length, as illustrated in the figure below for the case $n=5$. In this diagram, a sub-triangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid-lines.
It is desired to colour each vertex of the small triangles either red or blue in such a way that there is no sub-triangle with all three of its vertices having the same colour. Let $f(n)$ denote the number of distinct colourings satisfying this condition.
Determine, with proof, $f(n)$ for every $n \ge 1$.
It is desired to colour each vertex of the small triangles either red or blue in such a way that there is no sub-triangle with all three of its vertices having the same colour. Let $f(n)$ denote the number of distinct colourings satisfying this condition.
Determine, with proof, $f(n)$ for every $n \ge 1$.