Fibonachi Episod 2 - Origin of Fibonacci Sequence

The Fibonacci sequence was known in Indian mathematics independently of the West, but scholars differ on the timing of its discovery. Susantha Goonatilake writes that "Its development is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c.1135 AD), and Hemachandra (c.1150)." Parmanand Singh cites Pingala's contributions to the analysis of prosody, but writes that Virahanka was "the first authority who explicitly gave the rule for the formation" of the Fibonacci numbers. In contrast, Rachel Hall only mentions Hemachandra among these authors as having worked with Fibonacci numbers; she claims that around 1150, Hemachandra noticed that the number of possible rhythms followed the Fibonacci sequence. According to Donald Knuth, the motivation for the Indian study of these numbers came from Sanskrit prosody, where long syllables have duration 2 and short syllables have duration 1. As Knuth observes, Pingala numbered the different sequences that may be formed by these syllables, and Kedara (8th century AD) gave procedures for generating them all, but the connection to the Fibonacci sequence comes more specifically from considering the sequences with a fixed total duration, for which Knuth cites the anonymous Prakrta Paingala (c. 1320 AD). The number of sequences with duration m is the Fibonacci number F_m + 1, and the techniques in Prakrta Paingala involve Fibonacci coding.
In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202). He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that: a newly-born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

• At the end of the first month, they mate, but there is still only 1 pair.
• At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
• At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
• At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n-2) plus the number of pairs alive last month. This is the nth Fibonacci number.






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