An Inverse Theorem on Fibonacci Numbers

Naotaka Imada

A chapter in Applications of Fibonacci Numbers, 1990, pp 171-179 from Springer

Abstract: Abstract Let us begin with the particular case. Fibonacci Sequence {F n } (n=0, 1, 2,…) is defined by the three-term linear recurrence formula (1.1) $${F_n} = {F_{n - 1}} + {F_{n - 2}}\left( {n \geqslant 2} \right),{F_0} = 0,{F_1} = 1.$$ . Whence we have the following well-known relation (1.2) $${F_{n - 1}}\left( {{F_n} + {F_{n + 1}}} \right) - {F_n}{F_{n + 1}} = {\left( { - 1} \right)^n}\left( {n \geqslant 1} \right).$$ . Conversely, under the additional condition $${F_0} = 0,{F_1} = {F_2} = 1,{F_3} = 2$$ we can derive (1.1) from (1.2) by induction.

Keywords: Complex Number; Additional Condition; Chebyshev Polynomial; FIBONACCI Number; Fibonacci Sequence (search for similar items in EconPapers)
Date: 1990
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DOI: 10.1007/978-94-009-1910-5_19

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