Fibonacci or Hemachandra Numbers

Manish at Sepia Mutiny has an interesting entry on Fibonacci numbers which in fact should be called Hemecandra numbers.

The Fibonacci series is the set of numbers beginning with 1, 1 where every number is the sum of the previous two numbers. The series begins with 1, 1, 2, 3, 5, 8, 13, and so on. They were known in India before Fibonacci as the Hemachandra numbers. And the ratio of any two successive Fibonacci numbers approximates a ratio, ~1.618, called the golden section or golden mean.

It’s long been known that the Fibonacci series turns up freqently in nature. The numbers of petals on a daisy and the dimensions of a section of a spiral nautilus shell are usually Fibonacci numbers. For plants, this is because the fractional part of the golden mean, a constant called phi (0.618), is the rotation fraction (222.5 degrees) which yields the most efficient and scalable packing of circular objects such as seeds, petals and leaves.

But Bhargava points out that the series also shows up in the arts. Sanksrit poetry, tabla compositions and tango, to name a few examples, use the series to find the number of possible combinations of single and double-length beats within a stanza.[Sepia Mutiny: Hemachandra numbers everywhere]

Fibonacci himself wrote that he had studied Indian numbers and did not come up with the number series. Donald Knuth also wrote about this

Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is Fn+1; therefore both Gospala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, … explicitly.[Who was Fibonacci?]

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