Binary signed-digit integers and the Stern diatomic sequence

Monroe, Laura

doi:10.1007/s10623-021-00903-6

Cited by 3 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first result of this paper is a weight-distribution theorem identifying the number of i-bit BSD representations of an integer n having weight i − with the coefficient of the th term of the Stern polynomial of 2 i − n. This refines the result in [9], and is the basis for the rest of the paper.…”

Section: Introductionmentioning

confidence: 58%

“…In [9], Monroe noted a correspondence between the BSD representations of n and the hyperbinary representations of 2 i −1−n. This correspondence is restated here in Theorem 4.…”

Section: The Weight Distribution Of the Bsd Representations Of An Int...mentioning

confidence: 99%

“…As noted in [9], Theorem 4 gives the following transform between the digits of a BSD representation (b i−1 • • • b 0 ) of an integer n and the digits of the corresponding hyperbinary representation, (…”

Section: The Weight Distribution Of the Bsd Representations Of An Int...mentioning

confidence: 99%

“…In [9], a correspondence was shown between the number of BSD representations of an integer and Stern's diatomic sequence. We extend that work here, and examine the relationship of the related Stern polynomial [7] to the number of integer BSD representations, and in particular to optimal representations.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Binary Signed-Digit Integers and the Stern Polynomial

Monroe¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in some depth. In this paper, we show previously unknown connections between BSD representations and the Stern polynomial.We derive a weight-distribution theorem for i-bit BSD representations of an integer n in terms of the coefficients and degrees of the terms of the Stern polynomial of 2 i − n.We then show new recursions on Stern polynomials, and from these and the weightdistribution theorem obtain similar BSD recursions and a fast O(n) algorithm that calculates the number and number of 0s of the optimal BSD representations of all of the integers of NAF-bitlength log(n) at once, which then may be compared.

show abstract

Section: Introductionmentioning

confidence: 58%

“…In [9], Monroe noted a correspondence between the BSD representations of n and the hyperbinary representations of 2 i −1−n. This correspondence is restated here in Theorem 4.…”

Section: The Weight Distribution Of the Bsd Representations Of An Int...mentioning

confidence: 99%

Section: The Weight Distribution Of the Bsd Representations Of An Int...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Binary Signed-Digit Integers and the Stern Polynomial

Monroe¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The first technique involves normalised additive factorisation , which has been used in the Stern–Brocot tree (Bates et al [2]). Normalised additive factorisation is a binary signed-digit representation of an integer (see Monroe [9], Ebeid and Hasan [6], Tůma and Vábek [13] and Shallit [11]). The second technique is a different binary signed-digit representation of an integer.…”

Section: Introductionmentioning

confidence: 99%