On the sum of the reciprocals of <i>k</i>-generalized Fibonacci numbers release_t34xwhi7tndm3jyyigjzfvntxi

by Adel Alahmadi, Florian Luca

Published in Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica by Walter de Gruyter GmbH.

2022   Volume 30, p31-42

Abstract

<jats:title>Abstract</jats:title> In this note, we that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2022-0002_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msub> <m:mrow> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mi>n</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>k</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msubsup> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:tex-math>{\left\{ {F_n^{\left( k \right)}} \right\}_{n \ge 0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the <jats:italic>k</jats:italic>-generalized Fibonacci sequence then for <jats:italic>n</jats:italic> ≥ 2 the closest integer to the reciprocal of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2022-0002_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msub> <m:mo>∑</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:msubsup> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mi>m</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>k</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msubsup> </m:mrow> </m:mrow> </m:math> <jats:tex-math>\sum\nolimits_{m \ge n} {1/F_m^{\left( k \right)}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2022-0002_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mi>n</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>k</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:msubsup> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>k</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msubsup> </m:mrow> </m:math> <jats:tex-math>F_n^{\left( k \right)} - F_{n - 1}^{\left( k \right)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.
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