Let us define $G$ as the sequence of non-negative integer numbers such that the terms of indexes 0, 1, 2 are the numbers 0, 1, 2 and every next term is defined as the smallest integer number larger than the last already defined term such that if the term we are looking for is $k$ there exist no four distinct terms of $G$ - $a, b, c, d$ such that $a + b + c = 3 \cdot d$ (implying $k$ is among $a,b,c$ as if $k=d$ it is quite obvious that the sequence is equivalent to all non-negative integers).
The first few terms are as follows: 0, 1, 2, 3, 4, 12, 13, 14, 15, 16, 48, ...
R. P. Stanley wrote in one of his papers that it can be shown that the members of this sequence only if their expansion in base 4 satisfies that only the last digit may equal 2 and that if a digit is 1 all that follow are 0.
Here is a link to the paper: http://www.dtc.umn.edu/~odlyzko/unpublished/greedy.sequence.pdf
I've wondered if an easy proof exists for this that someone is willing to explain. I've recently gotten into number sequences and am researching the topic just for fun (I'm in high school) and am relatively inexperienced. Any help will be appreciated.
Thanks in advance!
P.S: I'm also interested in general techniques I can use when I'm trying to prove something about a said expansion (in a given base) of the terms of a sequence.