New family of Fibonacci Sequences 1

The Fibonacci sequences has a long and widely known reputation. It grabs your attention for a reason, as it has tremendously exciting features. So it is not surprising that he also inspired various generalizations. There is a fascinating literature on the subject, and anyone can easily believe that here we cannot expect big news.

Such a statement would be very careless. We should know that mathematics will not run out, and certainly will never run out of surprises.

For some time now I have been dealing with some issues in the sequences, and in the meantime I came across a little surprise.

At first, looking at a special version of the Fibonacci sequences, I was struck by a dramatic change. Well, now is the time to revive the definition of the Fibonacci sequences for the sake of order and to record some notations.

The Fibonacci sequence is the sequence of numbers denoted by F(n) in natural numbers such that F(1) = 1, F(2) = 1, and (if n> 2) F(n) = F(n -2) + F(n-1). The essence of the definition is that the first two terms of the sequences are given separately and in advance, from then on each subsequent member is the sum of the previous two. Since this is a summation of numbers (not, for example, division), we cannot expect any "difficulty" in calculating the sequences.

The Fibonacci sequence has a long and widely known reputation. It grabs your attention for a reason, as it has tremendously exciting features. So it is not surprising that he also inspired various generalizations. There is a fascinating literature on the subject, and anyone can easily believe that here we cannot expect big news.

Such a statement would be very careless. We should know that mathematics will not run out, and certainly will never run out of surprises.
For some time now I have been dealing with some issues in the sequences, and in the meantime I came across a little surprise.

Immediately, the first two terms are very arbitrary, and this is where most of the generalization of the sequences comes from. That is, any sequences with a first member a and a second member b is considered a Fibonacci sequences (the other rule remains the same for the other members). This means that any pair of numbers (a,b) can be assigned to a distinct sequence, which is obviously denoted by F_a,b(n).

It is very easy to see some striking features, such as selecting two consecutive members from a given sequences and launching a new sequences with them is not really new at all, on the contrary the new sequences is completely identical to the original from the two selected numbers.

Among other things, this property made it necessary to examine the possible history of the sequence, ie the calculation of the sequence "backwards". This immediately led to the discovery of sequences of exciting new features and relationships, but again proved to be a generalization. Here the rule changed like this: F*(n) = F*(n-2) - F*(n-1).

Because subtraction is as much a hassle operation as addition, we immediately get an infinite sequence (meaning all integers) in both directions, which is the original Fibonacci sequences in one direction and the same in the other direction.

It is also worth noting that this "mixing" is already due to the choice of the first two members of the sequences (or at least one negative number (which is not prevented by the impetus of generalizations).

Whichever direction we look at these sequences now, there are some very interesting rules at the markings of the sequences: each sequences has a peculiar "breakpoint, and from then on the sequences is sharply increasing or sharply decreasing in one direction, and the signs in the other direction alternate regularly.

It was this sign change that led me to "tile a sign change into the basic formula itself, so I wrote this formula:

F^S(n)=((-1)^(F^S(n-2)+F^S(n-1)))* F^S(n-2)+F^S(n-1)

To my surprise, this sequence turned out to be periodic, and not only with the original 1, 1 starting members, but with other starting numbers in every subsequent attempt. This, in the case of a standard mathematical formula (using only basic operations as addition and subtraction, multiplication and division, power and root subtraction), to say the least, was surprising.

Here are some examples:

Interestingly, the last column shows that the sequence of numbers immediately becomes negative and thus, underwater, floats to the other side of the new cycle.

Yes, but some new experiments have brought to light the lability of this new feature, or rather the Achilles heel. What happens is that if both of the novice members - and only then - lose the periodic nature of the sequences, they will switch back to the "regular" Fibonacci sequences. However, it should be noted that it takes the multiplication with any number very easily, since ac + bc = (a + b)c. Therefore, every Fibonacci sequences is considered a "basic" sequences multiplied by the largest common divisor of a and b. So if the "basic", or "serious" sequence, the two starting members can't be even at the same time (since 2 is still part of the largest common divisor). So, in fact, such an unfortunate clone would be able to fall out of the periodic character. Anyway, it's good to know about this Achilles heel. Otherwise, it is not difficult to understand why this "setback" occurs in the classic form: it is simply that for a pair of even pairs, the sum of each of the two adjacent members of the sequences is even, which generically blocks (unifies) the (-1) prefix in the training rules of the sequences.

But even with this weird exception, the basic case is surprising and interesting. A number of things are easy and quick to determine, but there are likely to be other exciting questions to ask.

One of the most important statements is that the length of the period is constant and it is 6 members. It does not depend on the sign of the initial members, the size of the members, whether b is greater or lesser, or even equal to a.

It is also a constant property that the cycle changes signs at least twice, i.e. at least one member has a different sign than the other. However, as shown in the examples above, the sign change can span up to four members.

Leonardo Pisano (Fibonacci), 1175-1250

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