Incidentally, it was Édouard Lucas in the 19th century who gave the Fibonacci sequence that name: around that time, it was rarely known and when used was called by many names including “the sequence of Lamé” (as Gabriel Lamé had used the sequence for analysis of the Euclidean algorithm). It was Lucas who noticed that one of the many problems in Fibonacci’s book involved that sequence, and started calling it the Fibonacci sequence. He was the one who discovered many properties and thereby made the sequence a worthy object of study (and “Lucas sequence” is now used as a general term that includes the Fibonacci sequence but also many others).
For the history of this sequence in India (specifically, how they arose in the study of Sanskrit/Prakrit prosody), I had posted some pictures online a few years ago, from the respective pages of the original sources, e.g. here https://twitter.com/svat/status/868964211644284928 you can see red circles around the Sanskrit words for “1”, “2”, “3”, “5”, “8”, “13”, “21”.

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