MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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}$}?

share|cite|improve this question
up vote 4 down vote accepted

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).

share|cite|improve this answer
Many thanks for the reference. Best 9i – Jörg Neunhäuserer Feb 16 '13 at 15:39
No problem. Hope it'll help. – Nikita Sidorov Feb 16 '13 at 22:41

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.