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The first 6 numbers of the Fibonacci is sometimes written as:

0, 1, 1, 2, 3, 5 (fib1)
Or
1, 1, 2, 3, 5, 8 (fib2)

Then there are the Lucas numbers:

2, 1, 3, 4, 7, 11

However, what I am doing is starting the sequence counter at 0 and continuing where the first number (n) increments by 1 in each new sequence and the second number is the constant, always 1. For example:

0   1   1   2   3   5   8   13 (fib1)
1   1   2   3   5   8   13  21 (fib2)
2   1   3   4   7   11  18  29 (lucas)
3   1   4   5   9   14  23  37 (??)
4   1   5   6   11  17  28  45 (??)
5   1   6   7   13  20  33  53
6   1   7   8   15  23  38  61
7   1   8   9   17  26  43  69
8   1   9   10  19  29  48  77
9   1   10  11  21  32  53  85
10  1   11  12  23  35  58  93
...

NOTE: This is not the same as "n-step Fibonacci". As far as I can tell, these are just considered to be "Fibonacci-like integer sequences", just as Lucas numbers.

One property I am interested in is how some numbers, found in j[] (see below), will not show up until n = i - 2 (where i is a number in set j[]). I consider these numbers unique. You can see in the sample data that the numbers 4, 6, and 10 do not show up anywhere in the matrix until n is equal to 2, 4, and 8 respectively. I did a manual search through a matrix of these sequences that was generated until the right-most number in each sequence was greater than or equal to 100, and found the numbers in j[] that meet this condition through 100.

Numbers in j[]: 4 6 10 12 16 22 24 30 36 40 42 46 52 54 64 66 70 72 82 84 90 94 96 100

Is there a name for unique numbers such as these found in fib-like sequences? I'm trying to find existing research on this subject.

aiwyn
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1 Answers1

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What you need is a general equation that allows you to examine the kind of relationships you are looking for. The discussion below parameterizes the results for any generalized Fibonacci-type sequence in terms of the initial conditions.

There have been many extensions of the sequence with adjustable (integer) coefficients and different (integer) initial conditions, e.g., $f_n=af_{n-1}+bf_{n-2}$. (You can look up Pell, Jacobsthal, Lucas, Pell-Lucas, and Jacobsthal-Lucas sequences.) Maynard has extended the analysis to $a,b\in\mathbb{R}$, (Ref: Maynard, P. (2008), “Generalised Binet Formulae,” $Applied \ Probability \ Trust$; available at http://ms.appliedprobability.org/data/files/Articles%2040/40-3-2.pdf.)

We have extended Maynard's analysis to include arbitrary $f_0,f_1\in\mathbb{R}$. It is relatively straightforward to show that

$$f_n=\left(f_1-\frac{af_0}{2}\right) \frac{\alpha^n-\beta^n}{\alpha-\beta}+\frac{af_0}{2} \frac{\alpha^n+\beta^n}{\alpha+\beta}= \left(f_1-\frac{af_0}{2}\right)F_n+\frac{af_0}{2}L_n$$

where $\alpha,\beta=(a\pm\sqrt{a^2+4b})/2$, $F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$, and $L_n=\frac{\alpha^n+\beta^n}{\alpha+\beta}$.

The result is written in this form to underscore that it is the sum of a Fibonacci-type and Lucas-type Binet-like terms. It will also reduce to the standard Fibonacci and Lucas sequences for $a=b=1, f_1=1, \text{ and } f_0=0 \text{ or }2$.

This can also be expressed as

$$f_n=f_1F_n+bf_0F_{n-1}$$

Basically, there are too many possible solutions (an infinity, really) to go around naming them all. Moreover, this result is not limited to the natural numbers and can be analytically continued by substituting the complex variable $z$ for $n$.

Cye Waldman
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