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fibonacci

# fibonacci - Fibonacci sequences Jonathan L.F King...

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Fibonacci sequences Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 26 April, 2009 (at 12:08 ) Prolegomenon. The famous Fibonacci sequence ~ f := ( f n ) n = is defined by f 0 := 0, f 1 := 1 and f n +1 = f n + f n - 1 . 1 : Let P and G be the Positive and neGative roots of Fib( x ) := x 2 - x - 1 note ==== [ x - P ][ x - G ] . So P + G = 1 and P · G = 1 . Moreover, P 2 = P + 1 and G 2 = G + 1 . 2 : Let ρ := 5 and b := 1 5 . Certainly ~ f is some linear combination α · [ n 7→ P n ] + β · [ n 7→ G n ]. Thus n Z : f n = b · [ P n - G n ] , 3 : since this formula gives correct values for f 0 and f 1 . 4 : Theorem. For each integer N : [ f N ] 2 + [ f N - 1 ] 2 = f 2 N - 1 . 4a : Proof. Always, LhS(4 a ) is non-negative. And RhS(4 a ) is non-negative, even when N Z - , since f Odd is always non-negative. So ISTProve that the squares of LhS(4 a ) and RhS(4 a ) are equal. To this end, define R := ρ 2 · [ f 2 N - 1 ] note ==== ρ · [ P 2 N - 1 - G 2 N - 1 ] and L := ρ 2 · [ f N ] 2 + [ f N - 1 ] 2 .

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fibonacci - Fibonacci sequences Jonathan L.F King...

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