Abstract
This paper concerns the properties of the generalized bi-periodic Fibonacci numbers {Gn} generated from the recurrence relation: Gn=aGn−1+Gn−2 (n is even) or Gn=bGn−1+Gn−2 (n is odd). We derive general identities for the reciprocal sums of products of two generalized bi-periodic Fibonacci numbers. More precisely, we obtain formulas for the integer parts of the numbers ∑k=n∞(a/b)ξ(k+1)GkGk+m−1,m=0,2,4,⋯, and ∑k=n∞1GkGk+m−1,m=1,3,5,⋯.
Highlights
We extend the results in [17] by considering the reciprocal sums of products of two generalized bi-periodic Fibonacci numbers
This paper concerned the properties of the generalized bi-periodic Fibonacci numbers { Gn } generated from the recurrence relation: Gn = aGn−1 + Gn−2 (n is even) or
We derived quite a general identities related to reciprocal sums of products of two generalized bi-periodic Fibonacci numbers
Summary
N=0 = S ( G0 , G1 , a, b ) to denote the generalized bi-periodic Fibonacci numbers { Gn } generated from the recurrence relation aGn−1 + Gn−2 , if n ∈ Ne ; Gn = Ohtsuka and Nakamura [6] reported an interesting property of the Fibonacci numbers { Fn } = S(0, 1, 1, 1) and proved the following identities: censee MDPI, Basel, Switzerland. Reciprocal sums of the generalized bi-periodic numbers were considered in [16,17].
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