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## A Third Generalization (Part 1)Welcome again to the wonderful world of Fibonacci
mathematics. We will again derive formulae for rabbit
populations others than the Fibonacci one. But before doing
so let us start with the binomial summation formula for the
generalised Fibonacci sequence We consider now the sequence of every second element of
The recurrence and binomial summation formulae for these two sequences read We verify first the recurrence formulae and then show the binomial summation formulae fulfilling the recurrence formulae. Observing G ( n ) = G ( n-1 ) + G ( n -2 ) and G ( n-1 ) = G ( n -2 ) + G ( n-3 ) we obtain G ( n ) = 2 G ( n -2 ) + G ( n-3 ) and with G ( n-3 ) = G ( n-2 ) - G (n-4 ) G ( n ) = 3 G ( n-2 ) - G ( n-4 ) . Replacing n by 2n+3 finally we obtain G ( 2n+3 ) = 3 G ( 2n+1 ) - G ( 2n-1 ). Replacing n by 2n+4 we obtain G ( 2n+4 ) = G ( 2n+2 ) - G ( 2n ). For proof of the binomial summation expression for G ( 2n+3 ) we use Rewriting G ( 2n+3 ) we obtain On the other hand we have The proof for G ( 2n+4 ) is very similar to the one for G (2n+3 ). Using G ( 1 ) = F ( 1 ) = 1 and G ( 3 ) = F ( 3 ) = 2 we obtain Using G ( 2 ) = F ( 2 ) = 1 and G ( 4 ) = F ( 3 ) = 3 we obtain G ( 1 ) = L ( 1 ) = 1 , G ( 2 ) = L ( 2 ) = 3 , G ( 3 ) = L ( 3 ) = 4 and G ( 4 ) = L ( 4 ) = 7 yields the Lucas sequence We can replace n by 2n-1 and 2n respectively in our formula for G ( n ) yielding and by comparison (16, 17) Using G ( 1 ) = G ( 2 ) = 1 we obtain Summing up our formulae for G ( 2n-1 ) and G ( 2n ) over n observing ( 11 ) yields We rewrite In a very similar way we can derive the formula for
two very well known formulae ( 2 ). Below we present an example of Since at the beginning of this website we ask how we can determine the Fibonacci number for a given value of n without having to work all the terms of the preceding ones . We have an answer by our binomial summation formulae and with the respecticve Binet - formulae we have an additional possibility. These can be obtained by inserting into the respective Binet - formulae for G ( n ) the terms ( 2n-1 ) and ( 2n ) respectively yielding Observing We obtain for the summation in formulae The world of Fibonacci binomial sums is more many-coloured than you might suggest from the above presentations. In the second part we give some examples of alternative representations indeed without proof since this will be done for more general formulae. |