<abstract xmlns="http://www.w3.org/1999/xhtml">

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

In this note, we that if 


{ Fn(k) }n≥0
{\left\{ {F_n^{\left( k \right)}} \right\}_{n \ge 0}}

 denotes the k-generalized Fibonacci sequence then for n ≥ 2 the closest integer to the reciprocal of 


∑m≥n1/Fm(k)
\sum\nolimits_{m \ge n} {1/F_m^{\left( k \right)}}

 is 


Fn(k)−Fn−1(k)
F_n^{\left( k \right)} - F_{n - 1}^{\left( k \right)}

.

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

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the sum of the reciprocals of -generalized Fibonacci numbers

In this note, we that if 


{ Fn(k) }n≥0
{\left\{ {F_n^{\left( k \right)}} \right\}_{n \ge 0}}

 denotes the k-generalized Fibonacci sequence then for n ≥ 2...

On the sum of the reciprocals of k-generalized Fibonacci numbers

Publié en ligne: 12 mars 2022

Pages: 31 - 42

Reçu: 07 mai 2021

Accepté: 25 juil. 2021

DOI: https://doi.org/10.2478/auom-2022-0002

Mots clés
Linearly recurrent sequences

© 2022 Adel Alahmadi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the sum of the reciprocals of k-generalized Fibonacci numbers

Publié en ligne: 12 mars 2022

Pages: 31 - 42

Reçu: 07 mai 2021

Accepté: 25 juil. 2021

DOI: https://doi.org/10.2478/auom-2022-0002

Mots clésLinearly recurrent sequences

© 2022 Adel Alahmadi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Mots clés
Linearly recurrent sequences