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

We derive recurrence relations for the squares of the Horadam numbers <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2022-0016_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>w</mi></mrow><mi>n</mi><mn>2</mn></msubsup></mrow></math>
<tex-math>w_n^2</tex-math>
</alternatives>
</inline-formula>, where the Horadam sequence <italic>w</italic><italic>n</italic> is such that the numbers <italic>w</italic><italic>n</italic>, for <italic>n</italic> ∈ ℤ, are defined recursively by <italic>w</italic>0 = <italic>a</italic>, <italic>w</italic>1 = <italic>b</italic>, <italic>w</italic><italic>n</italic> = <italic>pw</italic><italic>n−</italic>1 <italic>− qw</italic><italic>n−</italic>2 (<italic>n ≥</italic> 2), where <italic>a</italic>, <italic>b</italic>, <italic>p</italic> and <italic>q</italic> are arbitrary complex numbers with <italic>p</italic> ≠ 0 and <italic>q</italic> ≠ 0. Some related results emanating from the recurrence relations such as reciprocal sums, partial sums, and sums with double binomial coefficients are also presented.
</abstract>

We derive recurrence relations for the squares of the Horadam numbers 


wn2
w_n^2

, where the Horadam sequence wn is such that the numbers wn, for n ∈ ℤ, are defined recursively by w0 = a, w1 = b, wn = pwn−1 − qwn−2 (n ≥ 2), where a, b, p and q are arbitrary complex numbers with p ≠ 0 and q ≠ 0. Some related results emanating from the recurrence relations such as reciprocal sums, partial sums, and sums with double binomial coefficients are also presented.

Recurrence Relations for the Squares of the Horadam Numbers and Some Associated Consequences

Vasyl Stefanyk Precarpathian National University

Tatra Mountains Mathematical Publications

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

We derive recurrence relations for the squares of the Horadam numbers 


wn2
w_n^2

, where the Horadam sequence wn is such that the numbers wn, for n ∈ ℤ, are...

Recurrence Relations for the Squares of the Horadam Numbers and Some Associated Consequences

Published Online: Feb 15, 2023

Page range: 17 - 28

Received: Mar 15, 2021

DOI: https://doi.org/10.2478/tmmp-2022-0016

Keywords
Horadam sequence, Fibonacci sequence, Chebyshev polynomial, reciprocal series, recurrence relation, partial sum

© 2022 Kunle Adegoke et al., published by Sciendo

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

Recurrence Relations for the Squares of the Horadam Numbers and Some Associated Consequences

Published Online: Feb 15, 2023

Page range: 17 - 28

Received: Mar 15, 2021

DOI: https://doi.org/10.2478/tmmp-2022-0016

KeywordsHoradam sequence, Fibonacci sequence, Chebyshev polynomial, reciprocal series, recurrence relation, partial sum

© 2022 Kunle Adegoke et al., published by Sciendo

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

Keywords
Horadam sequence, Fibonacci sequence, Chebyshev polynomial, reciprocal series, recurrence relation, partial sum