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Fibonacci Shmibonacci

[Disclaimer: I am not a mathematician - and I don't even play one on TV - so my terminology may not always be correct, but I try to get the nature of my observations across in "lay" terms; I think/hope I've done so. Of course, feel free to contact me with any corrections/refinements to my usage of mathematical(or pseudo-mathematical) terms.]

I'm putting this page on hold, i.e., making considerable revisions based on new information.
Ben Sparks has alerted me to the fact that at least some of my findings are well-known to mathematics after all: linear recurrence, characteristic equations, inter al. (I didn't come across this 'cuz, not being a mathematician, I was searching with the wrong terms and keywords.)
Until I can determine whether any of my findings are new, I'm no longer claiming to have discovered anything. (Being unfamiliar with higher math, this may take a while.)
Visitors to this page are still welcome read it, download the files, and comment - and if you're a mathematician and do find anything of note, I'd like to hear.

Yearses ago (before the Internet had good search capability), I made an aboservation related to the Fibonacci sequence, viz., not only are there an infinite number of sequences that lead to the calculation of φ (phi), the "golden ratio," but that the Fibonacci sequence (and its look-alikes, e.g., Lucas) can be used to calculate one root of the quadratic equation x² + x -1 = 0; and if you extend the series before 1 1 2 you can find the other root:

… -144 89 -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55 89 144  …

The ratio S_n/S_n-1 will be a root (read alternative quotient section for more info). The farther out you go in either direction, the closer the approximation will be. (Note that the same “direction” of numerator and denominator must be the same on both ends of the sequence).  For the above sequence and equation the roots are (approximately)  -144/89 (-1.61797753) & 89/144 (0.61805556) which are close to the actual roots using the quadratic formula.
[The equation that does provide the golden ratio (φ) using the quotient as described here is actually x² - x -1 = 0 with the sequence … 144 89 55 34 21 13 8 5 3 2 1 1 0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144  …   :: 144/89 (1.61797753) & 89/-144 (-0.61805556).

This can also be generalized to other quadratic equation: c₂x² + c₁x – c₀ = 0 with a sequence where c₂*S_n + c₁*S_n+1 + c₀*S_n+2 = 0_.

Furthermore, it doesn’t stop there: almost ANY polynomial equation of order P — c_px^p + c_p-1x^p-1 + c_p-2x^p-2 + ………… +c₂x² + c₁x¹ - c₀x⁰ = 0 — can be solved with a sequence (infinite in both directions) such that the sum of ((c_p*S_n) + (c_p-1*S_n+1) + (c_p-2*S_n+1) + … + (c₁*S_n+p-1) + (c₀*S_n+p)) = 0.

For years, I would search the Internet every once in a while to see whether these observations of mine had already been made, but it didn’t look as though it had.

When I recently came across Numberphile and Matt Parker's Stand-up Maths (with their seeming obsession with Fibonacci sequences and Fibonacci-wannabes) as well as other math videos, and none of them indicated knowledge of this correlation, I became more assured that I had made a discovery. Some of the YouTube videos sometimes came close, sometimes oh-so close, to this correlation but didn’t make the leap to the association between the equation and the sequence. [Videos I've viewed.]
As it turns out, at least some my observations are well-known mathematics, but as I research this, I am presenting my findings (in the hope that some of them are novel).

Observations (interesting & puzzling)

I've found some interesting and puzzling things about these sequences and associated equations:

Convergence (Rate & Failure to converge)

There are instances when the sequence does not seem to solve the associated polynomial equation, i.e., the quotient of adjacent numbers does not (or cannot) converge.

I've identified at least two patterns where this happens - hence the "almost" in my intro:

•  Sequence oscillates between positive and negative numbers, but not in alternating fashion, e.g., repetition of one positive number followed by two negative numbers - the values change.
  This seems to be indicative of an imaginary root. (Trying to create a sequence using imaginary or complex seed values doesn't help.)
• Sequence repeats the same numbers throughout.
  This seems to be related to missing internal terms in the equation, i.e., coefficient of 0.  (I'm not sure whether in these cases you can construct a non-repeating sequence.)
• Some sequences feature both patterns, when there is a missing term and a root is imaginary.

The rate at which a sequence converges to a solution depends on the relative magnitude of the coefficients: when the coefficients are of the same order of magnitude, the sequence will converge faster. (That's why the Fibonacci sequence converges so quickly - all coefficients of magnitude 1.) This is probably why equations with missing terms "misbehave;" the 0 coefficient(s) are of infinitely smaller magnitude than the other coefficient. [By changing the 0 to a very small number, the new factor will change the actual solution by a negligible amount, it will produce a well-behaved sequence and a solution very close - but convergence will be sloooooooooooooow.]

[As far as roots to equations higher than linear with imaginary components, I don't think creating sequences with complex numbers will help, as they really are just the addition of two sequences that fit the equation. See the section Sequence Addition.]

Sequence Addition
Adding the sequences for two different equations results in a sequence that is the solution for the product of the original equations. Each of the sequences will meet the requirements for the new equation, although the new sequence will not necessarily fit the lower order equations. [Note that in some cases, the resulting sequence will be one that does not converge.]

e.g.

One implication is that any sequence - even one that fits a lesser order, even a linear, equation - that fits a polynomial equation will provide a solution. Try the sequences in the above tables with the 4^th-order equation created.
[If you create sequences for linear equations of the solutions to x² + x -1 = 0, viz. ((1+sqrt(5))/2)x -1 = 0 or (1-sqrt(5))/2)x -1 = 0), they will also fit both the quadratic and 4^th power equations: e.g. …  1 -0.618 0.382 -0.236 0.146  -0.090 0.058 -0.034 0.021 ...   (I've used decimal approximations, but you can just as well use actual mathematical notation, with the actual quadratic formula, to create the sequence; it just gets messy, having to do division with squares of the formula which contains sqrt(5).)]

Transforming a Sequence
Taking the difference between adjacent numbers creates a sequence that corresponds to the same equation. Doing this sometimes is necessary for some equations that have real solutions but the original sequence results in division by 0, as in above example:

x² - 1 = 0: … 2, 0, 2, 0, 2, … (#DIV/0!) ⇒ … 2, -2, 2, -2 … (tada!, sequence still fits the equation, but no division problems)
(This observation is not reflected in the main spreadsheet.)

Roots & Quotients
Taking the quotient of two numbers n "spaces" apart then taking the n^th root will give the same result. (In any case, S_n/S_n+1 is the first root.)
E.g., for Fibonacci sequence: (21/89)^1/3 = 0.61805556.
Take care in the choice of roots when the sequence involves negative numbers.

Extended Quotients
Using the sum of x numbers, instead of just the two adjacent numbers, yields the same result.
E.g., for Fibonacci sequence: (21 + 34 + 55) / (34 + 55+ 89) = 0.61805556

[I think it is beneficial to use the sums of x numbers for polynomial of the x^th power; it may mitigate some instances of non-convergence, though I need to do additional work on this.]

Tree diagrams
In addition to simple number sequences, I am looking into tree diagrams, where different node types transform into a new set of node(s). From the little work I've done, I sense a relationship to these sequences, but I need to do more before I can tell how true that is.
(I took this direction after reading about Fibonacci rabbits. I figured that if the Fibonacci sequence corresponds to a tree, other polynomials should, too.)

Below are links to files that illustrate the concepts on this page:

W Sequences Sequence Examples Sequence Addition (preliminary)

Tree Diagrams for Sequences Tree Diagram Examples

[I'm working on a spreadsheet that will be more flexible and allow a wider range of polynomials. (The current spreadsheets have sheets that are fixed for equations up to 5th power.)]

For further Investigation

Quest for Full Solutions
Since a sequence with only two ends can seemingly provide only two solutions, I’m still looking for a procedure that will find all solutions to higher order equations.
I'm particularly interested in how to construct the sequence - i.e., what "seed values" to choose - to ensure that a real solution will be found if there is one.

When Sequences Fail (a/k/a misbehaving sequences)
So far, I've identified two situations in which the quotient of two numbers does not compute a solution to the equation correlated to a sequence:

• solutions with imaginary/complex components:
  indications: sequence looks well-behaved, but one or both "sides" of the sequence may devolve into sequences where the values change, but positive and negative numbers do not alternate "nicely" e.g., repetition of one positive/number followed by two negative/positive numbers.
  future investigation: Currently I cannot see any method which will convert the misbehaving end(s) into the imaginary/complex solutions, but I'm still trying.
   
• equations with missing internal terms:
  indications: 1) sequence may look well-behaved, but the quotients of succeeding number pairs do not converge, generally alternating between two values.
                     2) sequence does not look well-behaved at all; quotients do not converge as in (1).
  workarounds: 1) sometimes taking the square root of numbers two spaces apart provides a solution. (I'm looking into the possibility that the root may depend on the order of the equation.)
                        2) As discussed above, if you are looking to get a solution, modifying the equation by using a small coefficient instead of 0 will create well-behaved sequence, but the convergence will be very very slow. The computed solution will  be a very good approximation, nonetheless.
  gut feeling: I'm leaning to the conclusion that it is not possible to create a well-behaving sequence for this type of equation.
  future investigation: I'm looking to the specific conditions, e.g., does this happen only when all internal terms are missing, or even when some are missing.

I’m not sure whether either or both of these is a matter of manipulating a single sequence in a new way or finding a different sequence, e.g., by using different “seed” values.