Abstract
We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers. We give a graph interpretation of these numbers with respect to a special 3-edge colouring of the graph.
Highlights
The Fibonacci sequence is defined by the following recurrence relation Fn = Fn−1 + Fn−2 for n ≥ 2 with F0 = F1 =1
We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers
Among sequences of the Fibonacci type there is the Pell sequence defined by Pn = 2Pn−1 +Pn−2 for n ≥ 2 with the initial conditions P0 = 0 and P1 = 1
Summary
On k-Distance Pell Numbers in 3-Edge-Coloured Graphs If k = 2 from Theorem 8 we obtain the following formula for the classical Pell numbers and companion Pell numbers: Pn = 5Pn−2 + 2Pn−3 , Qn = 5Qn−2 + 2Qn−3 , n ≥ 3. We give a graph interpretation of the kdistance Pell numbers with respect to special edge colouring of a graph.
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