Let $F\_{i}$ be the fibonacci or a multinacci sequence. The number of representations of $N$ in the form
$ N=\sum\_{i=0}^{k}s\_{i}F\_{i}, s\_{i}\in ${0,1} 
is known. 


My question is what is known about sequence-based numeration systems given by other linear recurrences. 


To make the question precise,
i am interested in the recurrence 
$ G\_{i+4}=G\_{i+3}+G\_{i+2}+G\_{i+1}-G\_{i}$
with $G\_{0}=1$, $G\_{1}=2$, $G\_{2}=4$, $G(3)=8$. 


What is known about
$ \sharp\_{G} N:=${$(s\_{0},\dots,s\_{k})\in${0,1}$^{k+1}|N=\sum\_{i=0}^{k}s\_{i}G\_{i}$}?



Linear numeration systems
Some results on the quantity in question can be found in 


*J. M. Dumont, N. Sidorov and A. Thomas, Number of representations related to a linear recurrent basis, Acta Arithmetica 88 (1999), 371-394.*


We are mainly interested in the summatory function but there are also some upper bounds for the quantity itself. Our main assumption is that the corresponding root (of $x^4=x^3+x^2+x-1$ in your case) is a Perron number (in your example it's even Salem, so our results apply). 


