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- W4225087137 abstract "In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $left(G_nright)_{n geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $mathcal{G}_{G_0, G_1}^2!(k)$, we prove $mathcal{G}_{G_0, G_1}^2!(k) = gcdleft(G_k G_{k+1} - G_0 G_1,; G_{k+1}^2 - G_1^2,; G_{k+2}^2 - G_2^2right)$. Moreover, we provide very tantalizing closed forms in the specific settings of the Fibonacci, Lucas, and generalized Fibonacci numbers. We close with a number of open questions for further research." @default.
- W4225087137 created "2022-04-30" @default.
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- W4225087137 date "2022-04-26" @default.
- W4225087137 modified "2023-10-16" @default.
- W4225087137 title "GCD of sums of $k$ consecutive squares of generalized Fibonacci numbers" @default.
- W4225087137 doi "https://doi.org/10.48550/arxiv.2204.12626" @default.
- W4225087137 hasPublicationYear "2022" @default.
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