汇总折纸序列

Pub Date : 2024-03-25 DOI:10.1017/s0004972724000169

MARTIN BUNDER, BRUCE BATES, STEPHEN ARNOLD

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引用次数: 0

摘要

序列 $a( 1) ,a( 2) ,a( 3) ,\ldots, $ 在《整数序列在线百科全书》中标为 A088431, 其定义如下： $a( n) $ 是 $$ \begin{align*} 的续分数展开式中 $( n+1) $ 第三项分量的一半，即 $( n+2) $ 第三项。\sum_{k=0}^{\infty }\frac{1}{2^{2^{k}}}.\end{align*}$$ Dimitri Hendriks 认为它是折纸序列 A014577 的运行长度序列。本文证明了这个求和折纸序列的几个结果，并证实了亨德里克斯的猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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THE SUMMED PAPERFOLDING SEQUENCE

The sequence

$a( 1) ,a( 2) ,a( 3) ,\ldots, $

labelled A088431 in the Online Encyclopedia of Integer Sequences, is defined by:

$a( n) $

is half of the

$( n+1) $

th component, that is, the

$( n+2) $

th term, of the continued fraction expansion of

$$ \begin{align*} \sum_{k=0}^{\infty }\frac{1}{2^{2^{k}}}. \end{align*} $$

Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding sequence, A014577. This paper proves several results for this summed paperfolding sequence and confirms Hendriks’s conjecture.

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