On the sum of the reciprocals of $$\pmb {k}$$ k -generalized Pell numbers

Benedict Vasco Normenyo ()
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Benedict Vasco Normenyo: University of Ghana

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 163-174

Abstract: Abstract In this paper, we establish a formula for finding the closest integer to the reciprocal of $$\sum _{m\ge n}1/P_m^{(k)}$$ ∑ m ≥ n 1 / P m ( k ) where $$\{P_n^{(k)}\}_{n\ge -(k-2)}$$ { P n ( k ) } n ≥ - ( k - 2 ) is the k-generalized Pell sequence or the k-Pell sequence given by $$\begin{aligned} P_n^{(k)}=2P_{n-1}^{(k)}+\cdots +P_{n-k}^{(k)}\;\; \text { for all } n\ge 2, \end{aligned}$$ P n ( k ) = 2 P n - 1 ( k ) + ⋯ + P n - k ( k ) for all n ≥ 2 , with initial conditions $$P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdots =P_{0}^{(k)}=0$$ P - ( k - 2 ) ( k ) = P - ( k - 3 ) ( k ) = ⋯ = P 0 ( k ) = 0 and $$P_1^{(k)}=1$$ P 1 ( k ) = 1 . We show that the closest integer to the reciprocal of $$\sum _{m\ge n}1/P_m^{(k)}$$ ∑ m ≥ n 1 / P m ( k ) is given by $$P_n^{(k)}-P_{n-1}^{(k)}$$ P n ( k ) - P n - 1 ( k ) for every $$k\ge 2$$ k ≥ 2 and for every $$n\ge 2$$ n ≥ 2 .

Keywords: Linearly recurrent sequences; k-generalized Pell numbers; k-generalized Fibonacci numbers; Closest integer; Reciprocal sum; 11B37; 11B39 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00463-6

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