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Thomas Baruchel

Publications and source records attributed to Thomas Baruchel.

7 recordsLinked to original sources
arXiv2025

An Efficient Algorithm for Permutation Iteration Using a Singly Linked List

We present a new content-agnostic algorithm for iterating over all permutations of a sequence. The algorithm leverages elementary~$O(1)$ operations on recursive lists. As a result, no new nodes are allocated during the computation. Instead, all elements are rearranged within the original nodes of the singly linked list throughout the process. While permutations are generated in an unusual order, the transitions between consecutive permutations remain smooth. A proof of concept written in the Lisp programming language is referenced and discussed.

cs.DS↗
arXiv2020

A non-symmetric divide-and-conquer recursive formula for the convolution of polynomials and power series

Some changes in a recent convolution formula are performed here in order to clean it up by using more conventional notations and by making use of more referrenced and documented components (namely Sierpiński's polynomials, the Thue-Morse sequence, the binomial modulo~2 transform and its inverse). Several variants are published here, by reading afterwards summed coefficients in another order; the last formula is then turned back from a summation to a new divide-and-conquer recursive formula.

math.NT↗
arXiv2019

Flattening Karatsuba's recursion tree into a single summation

The recursion tree resulting from Karatsuba's formula is built here by using an interleaved splitting scheme rather than the traditional left/right one. This allows an easier access to the nodes of the tree and $2n-1$ of them are initially flattened all at once into a single recursive formula. The whole tree is then flattened further into a convolution formula involving less elementary multiplications than the usual Cauchy product. Unlike the traditional splitting scheme, the interleaved approach may also be applied to infinite power series, and corresponding formulas are also given.

math.NT↗
arXiv2019

Properties of the cumulated deficient binary digit sum

The sequence A268289 from the On-Line Encyclopedia of Integer Sequences, namely the cumulated differences between the number of digits 1 and the number of digits 0 in the binary expansion of consecutive integers, is studied here. This sequence happens to match a sequence of cardinalities of some specific sets. Furthermore, it can also be expressed by using the Takagi function. The three different definitions have their own properties and combining them together lead to some new identities.

math.NT↗
arXiv2016

On error sums formed by rational approximations with split denominators

In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_mα- \frac{a_m}{c_m}\,\Big) \,,\] where $α$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m ={(-1)}^m$. In particular, we investigate such sums for \[α\in \big\{ π, e,e^{1/2},e^{1/3},\dots, \log (1+t), ζ(2), ζ(3) \big\} \] and exhibit some connections between rational coefficients occurring in error sums for Apéry's continued fraction for $ζ(2)$ and well-known integer sequences. The concept of the paper generalizes the theory of ordinary error sums, which are given by $b_m=q_m$ and $a_m/c_m=p_m$ with the convergents $p_m/q_m$ from the continued fraction expansion of $α$.

math.NT↗
arXiv2014

Variations on an error sum function for the convergents of some powers of $e$

Several years ago the second author playing with different "recognizers of real constants", e.g., the LLL algorithm, the Plouffe inverter, etc. found empirically the following formula. Let $p_n/q_n$ denote the $n$th convergent of the continued fraction of the constant $e$, then $$ \sum_{n \geq 0} |q_n e - p_n| = \frac{e}{4} \left(- 1 + 10 \sum_{n \geq 0} \frac{(-1)^n}{(n+1)! (2n^2 + 7n + 3)}\right). $$ The purpose of the present paper is to prove this formula and to give similar formulas for some powers of $e$.

math.NT↗