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## A Sixth GeneralizationWe continue in the following page with a binomial summation formula of the more general linear second order recurrence equation We proof by induction writing A Obviously A We assume A Then we state A This is just our statement when replacing n by r + 2. So our binomial summation formula holds for any r. From (70) some other formulae emerge. A linear second order recurrence formula in conjunction with
a binomial summation formula for a serie composed of every k-th element of the generalized Fibonacci serie
G We proof the recurrence formula by using Binets formula for G And by rearranging the above binomial summation formula we obtain finally We consider now the more general case A We proof first (75) in a very similar way then (71) Observing (69) and (70) yields (76). In particular when m = p = 1 we obtain formulae (71) and (72). In our next page we are concerned with linear third order recurrence formulae, A recurrence formula A has often been called a “Tribonacci“ sequence. We are concerned there with a sequence
composed of the squares of F |