The Fibonacci Numbers can be defined in different ways.

$$F_0 = 0; F_1 = 1$$

$$F_0 = 1; F_1 = 1$$

It doesn’t really matter.

However, the tag wiki says unequivocally that $F_0 = 1$, while the excerpt says equally firmly that $F_0 = 0$. Should both wiki and excerpt be edited to be a little more open to variation?

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    $\begingroup$ There are many properties of the Fibonacci numbers that rely on $F_0=0$ such as $F_k\mid F_{kn}$. $\endgroup$ – robjohn Dec 23 '14 at 14:02
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    $\begingroup$ It is better to have $F_0$ defined as zero in order to have $\gcd(F_n,F_m)=F_{\gcd(n,m)}$ and other nice identities. $\endgroup$ – Jack D'Aurizio Dec 23 '14 at 14:02
  • $\begingroup$ Integer sequences gives the sequence of Fibonacci numbers as $(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,\cdots)$. $\endgroup$ – Dietrich Burde Dec 23 '14 at 14:13
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    $\begingroup$ Why this was migrated, actually? This question is about editing tag wiki. $\endgroup$ – Grigory M Dec 23 '14 at 14:18
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    $\begingroup$ You're right, @GrigoryM. I've flagged it asking for a mod to move it back to meta. $\endgroup$ – TRiG Dec 23 '14 at 14:23
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    $\begingroup$ (about tag wiki excerpt contradicting tag wiki) $\endgroup$ – Grigory M Dec 23 '14 at 14:23
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    $\begingroup$ While technically not the proper name, I do think that allowing variations is natural and good. $\endgroup$ – user173897 Dec 23 '14 at 15:32
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    $\begingroup$ Well, it looks like I picked up 20 rep in that weird time when this question moved meta -> main -> meta. $\endgroup$ – TRiG Dec 23 '14 at 17:22
  • $\begingroup$ When I migrated this from meta to math, the title was "Where do the Fibonacci numbers begin?" There was a link to the tag-wiki, but it was not a meta question when it was migrated to math. $\endgroup$ – robjohn Dec 23 '14 at 18:56
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    $\begingroup$ I invite you to reread the actual body of the question, @robjohn. $\endgroup$ – TRiG Dec 23 '14 at 20:44
  • $\begingroup$ I would avoid putting alternate definitions in the tag-wiki. Choose one and stick to it. It's not like someone's not going to ask a question because the tag has a shifted initial condition - in all likelyhood, they read the tag-wiki after posting the question (if at all). (Hey, even stick $F_0=0$, $F_1=1$ and $F_2=1$ for clarity to anyone who's not aware of the shift, if you want) $\endgroup$ – Milo Brandt Dec 24 '14 at 1:51
  • $\begingroup$ I think Fibonacci numbers deserve their own tag, since they are of significant interest independently of other related sequences. But perhaps questions about other sequences (Lucas numbers, etc.) deserve a general tag of their own. The [sequences-and-series] tag is overused in my opinion; it draws questions from calculus (convergence and summability) and combinatorics/number theory that have little to do with one another. I suggest the creation of an [integer-sequences] tag (I was sure there would be one!) to cover the latter case. $\endgroup$ – Gyu Eun Lee Dec 25 '14 at 19:48
  • $\begingroup$ @neuguy Other sequences defined by recurrence relations naturally fall under recurrence-relations tag. So do the Fibonacci numbers, but they appear to be sufficiently famous to also have own tag. $\endgroup$ – user147263 Dec 25 '14 at 20:43
  • $\begingroup$ Motivation for such conventions is already discussed in prior main questions, e.g. here and here. $\endgroup$ – Bill Dubuque Jan 3 '15 at 19:41

I suggest that we adopt the convention that $F_0=0$ for both the tag wiki and summary, with a possible note in the wiki that $F_0=1$ is occasionally used as well.

Why? This is consistent with OEIS A000045 and it results in a nice form of Binet's formula (as opposed to a offset in the exponents). Other properties (as mentioned in the comments) rely on a starting value of $0$.

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    $\begingroup$ There is no advantage to having $F_0=1$. $\endgroup$ – Ali Caglayan Dec 24 '14 at 20:10
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    $\begingroup$ @Alizter If you define $F_n$ as the number of tilings of $1 \times n$ rectangle by $1 \times 1$ and $1 \times 2$ tiles (which seems like a pretty natural definition) then you get $F_0 = F_1 = 1$. But I think it is obvious on the whole that $F_0 = 0$ is better. $\endgroup$ – 6005 Jan 3 '15 at 19:55

I suggest describing something like as follows:

Fibonacci Numbers are consecutive elements of the Fibonacci Sequence: $F_0=0, F_1=1, F_{n+1}=F_n+F_{n-1} (n\gt 0)$

ie distinguishing Fibonacci Numbers, where the indexing is irrelevant, from the Fibonacci Sequence, where canonical indexing is expected.


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