Matra vrttas and Fibonacci series

The sequence 1,1,2.3.5.8,13……… in which nth term is given by U=U+ U is known as Fibonacci series .Leonardo of Pisa (1170-1250) was known as Fibonacci. But the sequence was well known in India before Leonardo’s time. The Indian experts on metrical sciences used this sequence in work on metric.The basic unit in Sanskrit prosody are a letter having a single matra (mora or a syllabic instant) calledlaghu(light) and that having two morae called guru(heavy).And their role in metric is the same as that of 1 and 2 in combinatorics. Matra-vrttas are meters in which numbers of morae remains constant and the number of letters is arbitrary. The numbers of variations of meters having 1,2,3,4,5,6…………….morae are 1,2,3,5,8,13……………….respectively and these are the Fibonacci numbers. Acarya Pingala is the first authority on the metrical sciences in India. Pingala’s rule misrau ca (ie; and the two mixed) is also meant for the expansion of matra vrttas and matra vrttas should be expressed by combining the expansions of two earlier meters with a guru andlaghu respectively. Acarya Virahanka (between AD 600-800) given direction for knowing the number (of variations) of the next matra(vrtta). Gopala (prior to AD 1135) is the first author on metric who specifically mentions the numbers of variations of matra vrttas (known as Fibonacci numbers) while commenting on this rule of Acharya Virahanka. Acarya Hemachandra(AD 1150) ,one of the greatest Jain writers is also an authority on metric,who specifically mentions the number of variations of matra vrttas in hisChandonusasana. Acarya Hemachandra’s rule may be translated as follows: “ Sum of last and the last but one numbers(of variations) is that of the matra vrtta coming next.” Hemachandra concludes his statement “1,2,3,5,8,13,21,34 and in this way, afterwards”. Ganitha kaumudi(GK) written by Narayana Pandita in AD 1356 was the first mathematical work in which the idea of the so called Fibonacci number were developed further Chapter 13 of GK defines Samasika Pankti(additive sequence) .The rule may be translated as “First Keeping unity twice ,write their sum ahead, write ahead of that ,the sum of numbers from the reverse order(and in) place equal to the greatest digit, write ahead the sum of those (in available places).Numbers at places (equal to) one more than ‘the sum of digits’ happen to be Samasika Pankti The relation between the so called Fibonacci numbers and the binomial coefficients as contained in the prakrta pingala was established with the help of the matra meru(figure of numbers). Narayana Pandita established a relation between his samasika pankit and the multinomial coefficients