On the sum of the reciprocals of <i>k</i>-generalized Fibonacci numbers
release_t34xwhi7tndm3jyyigjzfvntxi
by
Adel Alahmadi,
Florian Luca
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>.
In application/xml+jats
format
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