In mathematics, the Fibonacci numbers, named after Leonardo of Pisa, known as Fibonacci, form a sequence defined recursively by:

$$

F_n := F(n):= \begin{cases} 0 & \mbox{if } n = 0; \\ 1 & \mbox{if } n = 1; \\ F(n-1)+F(n-2) & \mbox{if } n > 1. \\ \end{cases} In other words, after two starting values, each number is the sum of the two numbers before it. The first Fibonacci numbers for n = 0, 1, … are

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, …

Origins

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The "Fibonacci" numbers first appear, under the name maatraameru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chhandah-shāstra, the Art of Prosody, 450 or 200 BC). The Indian mathematician Virahanka gave explicit rules for the Fibonacci sequence in the 8th century. The Indian Jain philosopher Hemachandra (c.1150) (and also Gopala) revisited the problem in some detail. Sanskrit vowel sounds can be long (L) or short (S), and Hemachandra wished to compute how many cadences of a given overall length can be composed of these. If the long syllable is twice as long as the short, the solutions are:

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