In the Fibonacci sequence, is fib(0) 0 or 1 ?

Question

I'm doing a task in a subject were fib(0) is defined to = 1. But that can't be right? fib(0) is 0?

Program with fib(0) = 1; spits out fib(4) = 5 Program with fib(0) = 0; spits out fib(3) = 3 

What is the correct definition?

Answer 1

The definition with Fib(0) = 1 is known as the combinatorial definition, and Fib(0) = 0 is the classical definition. Both are used in the Fibonacci Quarterly, though authors that use the combinatorial definition need to add a sentence of explanation. Benjamin and Quinn in Proofs that Really Count use f_n for the nth combinatorial Fibonacci number and F_n for the nth classical Fibonacci number. The combinatorial definition is good, not surprisingly for counting questions like "How many ways are there to walk up a flight of n steps, taking either one or two steps at a time?" When n is 0, there's one way to do it, not zero ways.

Answer 2

You're correct. The Fibonacci sequence is formally defined with seed values fib(0) = 0 and fib(1) = 1. This is a requirement for the rest of the sequence to be right (and not offset by one or anything).

In mathematics, the Fibonacci numbers, commonly denoted F_n, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1.

In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1.

Edit: I have to concede that there is another (much less common, and usually informal) way to define the sequence by seeding it with values 1 and 1, but this is not the conventional one by any means. It is certainly not preferred in all the formal mathematical definitions I’ve seen, like The On-Line Encyclopaedia of Integer Sequences.