9 research outputs found
Non-standard binary representations and the Stern sequence
We show that the number of short binary signed-digit representations of an
integer is equal to the -th term in the Stern sequence. Various proofs
are provided, including direct, bijective, and generating function proofs. We
also show that this result can be derived from recent work of Monroe on binary
signed-digit representations of a fixed length
On certain arithmetic properties of Stern polynomials
We prove several theorems concerning arithmetic properties of Stern
polynomials defined in the following way: , and . We study also the
sequence e(n)=\op{deg}_{t}B_{n}(t) and give various of its properties.Comment: 20 page
Arithmetic properties of the sequence of degrees of Stern polynomials and related results
Let be a -th Stern polynomial and let e(n)=\op{deg}B_{n}(t)
be its degree. In this note we continue our study started in \cite{Ul} of the
arithmetic properties of the sequence of Stern polynomials and the sequence
. We also study the sequence
d(n)=\op{ord}_{t=0}B_{n}(t). Among other things we prove that ,
where is the maximal power of 2 which dividies the number . We also
count the number of the solutions of the equations and
in the interval . We also obtain an interesting closed expression
for a certain sum involving Stern polynomials.Comment: 16 page
Exact arithmetic on the Stern–Brocot tree
AbstractIn this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials