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I have searched the web for a definition and I found that this are typically defined as

$F^n_{m} = F^n_{m-1} + F^n_{m-2} + \dots + F^n_{m-n}$

where $n$ stands for the number of previous numbers who are added up, and $m$ the mth number who has to be obtained, with $m \geq n+1$. But the first nth numbers are defined as follows:

$F^n_{1}=F^{n-1}_{1}, F^n_{2}= F^{n-1}_{2}, \dots, F^n_{n}=F^{n-1}_{n}$

That is, you depend on the Fibonacci $n-1$ step numbers, so for example, the first numbers of the Fibonacci $3$ step numbers are:

$1,1,2,4,7, \dots$

for $n=4$:

$1,1,2,4,8,15,\dots$

You see the pattern. So my question is: Why the first nth numbers not defined as $n$ ones? that is:

$F^n_{1}=F^n_{2}=\dots=F^n_{n}=1$

And also How different results does this definition could give, compared to the previous one?

Rogelio
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  • The Fibonacci numbers aspire to be powers of two, and as such want to be the sum of all previous terms. However, they are nearsighted, and therefore cannot see all the way back to the beginning of the sequence unless they are sufficiently close. (Jokes aside, I think something like that is the real reason. Not the nearsightedness, but the homogeny of it, trying to make the initial terms as similar to the rest of the sequence as possible.) – Arthur Mar 02 '16 at 18:43

1 Answers1

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This definition is what you get if you define $F^n_0=F^n_{-1}=F^n_{-2}=...=0$.
Your definition is smaller by a factor that is smaller than $2^{n-1}$, but I don't know exactly.

Empy2
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