This analysis reveals identities involving lucas numbers and binomial coefficients, suggesting simple mathematical methods may simplify proofs.
We find and prove a number of simple connections between Lucas numbers and sums of binomial coefficients. We show how mathematical induction and elementary ideas can be used to obtain these identities, while minimizing the number of necessary computations. Some knowledge of simple properties of binomial coefficients, as well as a basic understanding of the Lucas and Fibonacci numbers, is all that is needed.
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Russell A. Gordon (2025) studied this question.
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