Little Prince

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  • Topic: Fibonacci number, Fibonacci, Golden ratio
  • Pages : 2 (553 words )
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  • Published : March 14, 2013
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The plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). A fractal-like series of white triangles appears on the bottom edge, due in part to the fact that the binary representation of  ends in  zeros. Many other similar properties exist. The Fibonacci numbers give the number of pairs of rabbits  months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa (also known as Fibonacci) in his book Liber Abaci. Kepler also described the Fibonacci numbers (Kepler 1966; Wells 1986, pp. 61-62 and 65). Before Fibonacci wrote his work, the Fibonacci numbers had already been discussed by Indian scholars such as Gopāla (before 1135) and Hemachandra (c. 1150) who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes or syllables. The number of such rhythms having  beats altogether is , and hence these scholars both mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly (Knuth 1997, p. 80). The numbers of Fibonacci numbers less than 10, , , ... are 6, 11, 16, 20, 25, 30, 35, 39, 44, ... (Sloane's A072353). For , 2, ..., the numbers of decimal digits in  are 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, ... (Sloane's A068070). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of  (Sloane's A097348), where  is the golden ratio. This follows from the fact that for any power function , the number of decimal digits for  is given by . The Fibonacci numbers , are squareful for , 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (Sloane's A037917) and squarefree for , 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (Sloane's A037918).  and for all , and there is at least one  such that ....
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