Fib sequence, is fib 0 = 0 or 1?

You're correct. The Fibonacci sequence is defined with seed values F_0 = 0 and F_1 = 1 This is a requirement for the rest of the sequence to be correct The only condition under which F_0 = 1 could work is if you defined a "-1 based counting system" (as opposed to the usual conventions of 0-based and 1-based). This would be pretty wacky however, I'm sure you agree.

You're correct. The Fibonacci sequence is defined with seed values F_0 = 0 and F_1 = 1. This is a requirement for the rest of the sequence to be correct.

The only condition under which F_0 = 1 could work is if you defined a "-1 based counting system" (as opposed to the usual conventions of 0-based and 1-based). This would be pretty wacky however, I'm sure you agree.

In other words, his sequence is offset by one index. – Markus Sep 20 '09 at 14:56 @Markus: Yes, offset in a very strange way. It could just be that whoever assigned the task got it wrong, however (more likely?).

– Noldorin Sep 20 '09 at 15:00 It's not a requirement, see the answer of Dale Gerdemann. – Sjoerd May 8 at 11:57 @Sjoerd: I've done enough mathematics to know it's simply non-standard. – Noldorin May 8 at 13:13.

Based on the definition of the Fibonacci sequence, you can generate a closed form for defining the nth element: F(n) = ( f^n - (1-f)^n ) / sqrt(5), where f = (1 + sqrt(5)) / 2 the golden ratio For n = 0 it is clearly 0: F(0) = (1 - 1) / sqrt(5) = 0.

That's an explanation, albeit a roundabout one. It's really the seed that defines it in the first place. – Noldorin Sep 20 '09 at 15:01 True, this is inside-out... – Zed Sep 20 '09 at 15:09 Anyway, there's certainly no debate on the closed form, so this gives an unquestionable answer to the question =) – Zed Sep 20 '09 at 15:12 2 @Noldorin Of course you could define the seed differently, but then a lot of nice theorems would become false, like this one.

BTW, my favorite is gcd(F_m, F_n) = F_gcd(m,n). – starblue Sep 20 '09 at 16:05.

From the Fibonacci number entry on Wikipedia: In mathematics, the Fibonacci numbers are the following sequence of numbers: By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s. In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation with seed values.

2 With a nice little emphasis on: "Some sources omit the initial 0, instead beginning the sequence with two 1s" – NomeN Sep 20 '09 at 15:41.

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. " When n is 0, there's one way to do it, not zero ways.

Fib 0 = 0 fib 1 = 1 That is the seed value definition.

Just stating that something is so, doesn't make it so. – Sjoerd May 8 at 11:58.

They are both correct. If you specify a sequence G{n} by the recursion G{1} = 3, G{2} = 5, G{n} = G{ n - 1} + G{ n - 2} then most people would agree that is "a Fibonacci sequence". The only difference being a few terms at the front, but the leading terms are mostly irrelevant for any interesting questions about the sequence.

The heart of a Fibonacci sequence is the addition rule, and any sequence that uses that rule is a Fibonacci sequence. It is only necessary to specify whether 0 is in the sequence if you want to ask specific questions about a particular index... every thing else is just a translation on the index and is pretty much irrelevant. That is, if the problem is 'find a closed form solution for the Nth value in the sequence', then solving it for G will solve the problem for F with just a trivial shift of the solution.

The hard part of the problem is the same for both sequences.

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