Journal of Integer Sequences, Vol. 27 (2024), Article 24.6.4

On k-Fibonacci Brousseau Sums


Prabha Sivaraman Nair
Department of Mathematics
Baby John Memorial Government College, Chavara
Kerala 691583
India

Rejikumar Karunakaran
Department of Mathematics
NSS College, Cherthala
Kerala 688541
India

Abstract:

Using elementary methods, we provide formulas for evaluating the Brousseau sum $\sum_{i=1}^{n} i^{p} F_{k,i}$ and the shifted Brousseau sum $\sum_{i=1}^{n} i^{p} F_{k,m+i}$ for all integers $m,p\geq0$, where $(F_{k,i})_{i\geq0}$ is the $k$-Fibonacci sequence defined by the two-term linear recurrence $F_{k,i}=kF_{k,i-1}+F_{k,i-2}$ for $i\geq2$ with initial values $F_{k,0}=0$ and $F_{k,1}=1$.

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(Concerned with sequences A000032 A000045 A000129 A000557 A001076 A002203 A002878 A006154 A006190 A077444 A259546.)

Received October 11 2023; revised versions received February 23 2024; July 17 2024. Published in Journal of Integer Sequences, July 17 2024.

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