The so-called Fibonacci Sequence is only one of an infinite family of sequences in which the ratio of the penultimate term to the ultimate term converges to the Golden Mean as the terms approach infinity; further, there is an infinity of similar families of sequences, each family converging to a unique ratio of penultimate to ultimate term.
Families of families, so to speak. Infinitely.
Further, there is a continuous function such that this result may be generalized for any combination of real numbers whatsoever.
The most interesting topic in Dan Brown’s The Da Vinci Code was the Fibonacci Sequence. The rule of the sequence is that each term in the sequence is the sum of the previous two terms. There is a primary sequence, familiar to almost everyone in numbers, which begins with the seed terms 1, 1 and proceeds to 2, 3, 5, 8, 13, 21, and so on, producing terms inexorably and endlessly.
The ratio xn/xn+1 converges upon the golden mean as the terms grow larger, where we take xn as the nth term and xn+1 as the next term after the nth term. (The exact value of the golden mean is (1/2) ((5)1/2 -1), and the decimal approximation of this value to eight places is .61803399.) The golden mean is the only numerical value which, when added to 1, forms its own reciprocal. We know of applications in architecture and biology.
I would think there are applications to time.
Since the ratio of any two successive terms in the sequence converges to a distinct value, one suspects a feedback system is at work, and that the golden mean may be considered an attractor. It is not the sort of attractor we are accustomed to, however, since the convergence requires a two-stage feedback.
The familiar or primary Fibonacci sequence is not the only sequence which converges to the Golden Mean. In fact any two numbers taken as seed terms (including the terms of the Lucas set, of course) will produce a sequence in which the ratio of xn to xn+1 converges to the golden mean. It does not matter whether the seed numbers are integral, rational, irrational, positive, or negative. In all cases the ratio of two successive terms converges to the golden mean as the absolute value of the terms grows larger. The number of terms added to produce the next term is what controls this convergence, not the numbers used as seeds.
Since in these instances 2 successive terms are added to produce the next term, it’s the number 2 which specifies the Golden Mean.
If you have a spreadsheet or a mathematics program, you may wish to test various Fibonacci sequences to satisfy yourself that no matter which two numbers are used as seeds, the ratio of xn to xn+1 converges to the Golden Mean.
The fact that the number 2 specifies, via Fibonacci-type sequences, the golden mean, raises the question whether other numbers might specify other values in a similar way. 3, for example. Or 4. Let us designate as a sum number a number which indicates how many successive terms of a sequence must be summed to produce the next term, and represent sum numbers by s. If we say that a sequence has a sum number of 3 (s = 3), we mean that 3 successive terms are summed to produce the next term. A sum number of 4 implies that 4 successive terms are summed to produce the next term, and so on. For every sum number s there is what I will call the primary sequence for that number, in which s successive values of 1 are summed to initiate the sequence—1, 1, 1, 1, 1, would be the seed numbers for the primary sequence in which s = 5, for example. Let us refer to all sequences generated by a given sum number s, including of course the primary sequence, as a family of sequences, S.
It develops, as one might expect from the behavior of the family of Fibonacci sequences (s = 2), that one may begin by summing any s numbers, and the ratio xn/xn+1 will converge (as the terms of the sequence grow larger) to a distinct value determined only by s.
For convenience let us refer to the limit of the ratio xn/xn+1 for any family of sequences S produced by a sum number s as rs.
There exists an infinite group of families of sequences, each family classifiable by the number of successive terms summed to produce the next term. I refer to the members of this general group as parafibs (para for “beyond” and “fib” for Fibonacci). All parafibs behave alike, including the Fibonacci sequence and its family, in that each family S converges to a ratio rs unique to that family.
Since every value of s correlates with one and only one value of rs, every positive sum number s implies one and only one other positive number, rs. Therefore s <=> rs, and we have a function, which we will designate as f(s). Is every rs implied by s irrational? It seems so, but I cannot think how to prove this result.
When the value of s is zero, no sequence can occur. We may designate this as the null sequence. To say s = 1 is to say that the sequence consists of the same number repeated endlessly and therefore that the ratio r1 = 1.
Can f(s) be made continuous—that is to say, is there a way to create fractional and irrational values for s which produce results consistent with the results of integral values?
The conceptual problem is how to define a portion of an operation. In this situation, it develops that there is one definition which suits our requirements. If we define s as v.w, where v is understood as the integral portion of s and .w is understood as the decimal portion, and we define a term xn in the sequence s as the sum of the (v -1) previous terms plus (.w) times the vth term, then we have a continuous function whose domain is between 0 and infinity.
It is notable that in this definition if 1 > rs > the golden mean, then rs = (j – 1)/j => that s = 1 + j/(j – 1)2 for all real values of j.
In the successful definition, if s < 1, the successive terms rapidly approach zero and rs = 1/s, so that rs approaches infinity as s approaches 0.
I’ve investigated other potential definitions, in particular one which reverses the method above, treating xn as the sum of (.w)(xn-1) plus the (v – 1) terms previous (xn-2, xn-3, . . . , xn-v) and one in which rv.w is defined as rv + (.w)(rv+1 – rv), and the results, while formally interesting, do not produce a continuous function over the interval.
We discover that when s is greater than 1, rs is less than 1 but greater than ½. In fact rs approaches ½ as s approaches positive infinity. I haven’t found any algebraic representation of f(s). Perhaps the function is transcendental.
Again, set up a spreadsheet or mathematical program to produce sequences with s-values of your choosing. See if you do not find, for example, that r2 is approximately equal to .54368901, and r12 to .50006113.
It develops that in any sequence s, 1/rs approaches 1+[xn-1+ . . . +xn-(s-1)]/xn as n approaches positive infinity. In the Fibonacci family, this formula reduces to the familiar reciprocal effect. (Since s= 2, xn-(s-1) = xn-1, and therefore 1/ rs = 1+xn-1/xn = xn/xn-1).
It’s interesting that any two successive terms (using forward slant bars to indicate absolute value) of any sequence from the Fibonacci family (xn, xn+1) are related by the equation /(xn)2 – ((xn-1)(xn+1))/ = c, in which equation c is a constant specific to that particular sequence. In the traditional sequence, for example, c = 1. In a sequence from the Fibonacci family whose seeds are 4 and 11, c = 61. (If one does not use the absolute value of (xn)2 – (xn-1)(xn+1), then c alternates between positive and negative values.) Similar results do not appear to apply to sequences whose sum number is not 2.
(Addendum: It seems clear to me, some time later, that what I have designated S ought to be designated S-sub-s, but I cannot find an editing tool on this site to generate subscripts, and while I am on this site, the formatting tool from the general overhead bar does not appear.)