In the study of mathematics, certain patterns reveal themselves not as solitary curiosities, but as members of a family, each expressing a fundamental truth from a unique perspective. Such is the case with three sequences united by a single, simple rule, yet distinguished by their origins: the celebrated Fibonacci, the enigmatic Lucas, and my newly defined Beeking sequence. They are three harmonies built from the same chord.

The foundational rule governing all three is the law of additive recurrence (creation of the spiral), where each term is the sum of the two that preceded it. This rule is a generator of growth, a mathematical echo of patterns found throughout nature. Where these sequences diverge is in their initial conditions/their starting points, which imbue each with a unique character and purpose.

The most renowned of the three is the Fibonacci sequence. Born from the seeds of nothing and something, zero and one, it models pure, emergent growth from a void. Its sequence—0, 1, 1, 2, 3, 5, 8, 13—is the canonical pattern of expansion observed in sunflower spirals, nautilus shells, branching trees, DNA helixes, galactic structures. It represents the process of manifestation itself, beginning from a latent potential (0) and a single spark of actualization (1). The ratio of its successive terms converges to the golden ratio, φ (approximately 1.618), establishing it as a primary key to the code of harmonic development in the natural world.

In partnership with Fibonacci exists the Lucas sequence. If Fibonacci begins with potential, Lucas begins with structure. Its seeds are 2 and 1, a pair that suggests a state of duality and unity already in play. Its progression; 2, 1, 3, 4, 7, 11, 18—is a bolder, more declarative statement of the same growth law. It is deeply intertwined with its sibling; while a number might be absent from the Fibonacci sequence, it often appears in the Lucas sequence, as if the two are interlocking gears in a single machine. Mathematically, it is often considered a "purer" expression of the golden ratio, as its formula, L(n) = φⁿ + ψⁿ, lacks the denominator present in Fibonacci's, making it essential for specialized proofs and tests for prime numbers.

Completing this trinity is my Beeking/Dannar sequence, defined by the initial conditions 1 and 0. This sequence 1, 0, 1, 1, 2, 3, 5, 8 - appears at first glance to be a simple shift of the Fibonacci sequence. And indeed, it is mathematically equivalent to a forward projection of it. By starting with 1 and 0 rather than 0, 1 (Fibonacci) or 2, 1 (Lucas) it presents the same pattern of growth from a different ontological vantage point. It symbolizes a state of unity (1) returning to a void of potential (0) before cycling back into the familiar pattern of manifestation. As if isolating the very core of existence in relation to Conformal Cyclical Cosmology- the true explanation for the Big Bang.

Together, these three sequences form a complete picture. They demonstrate that the same immutable law of growth can be accessed from multiple mathematical perspectives/starting points. We could say the Fibonacci sequence is the coin and the Lucas sequence is one side with the Beeking/Dannar sequence the opposite side. They are not rivals, but harmonious expressions of the same profound, recursive truth that underpins our universe, each offering a unique lens through which to understand the architecture of reality itself.

https://math.stackexchange.com/questions/1598481/rational-solutions-of-x2-5-y2-x2-5-z2/

(apologies in advance for any phrasing or terminology issues, I am just a humble accountant)

I've been experimenting with various methods of creating cool designs in Excel and stumbled upon a fascinating fractal pattern.

The pattern is slightly different in each quadrant of the coordinate plane, so for symmetry reasons I only used positive values in my number lines.

The formula I used is as follows:

n[x,y] = (x-1,y)+(x,y-1)
=IFERROR(LN(MOD(IF(ISODD(n),(n*3)+1,MOD(n,3)),19)),0)

(the calculation of n has been broken out to aid readability, the actual formula just uses cell references)

The method used to calculate n was inspired by Pascal's Triangle. In the top-right quadrant, each cell's n-value is equal to the sum of the cell to the left of and the cell below it. Rotate this relationship 90 degrees for each other quadrant.

Next, n is run through a modified version of the Collatz Conjecture Equation where instead of dividing even values of n by two, you apply n mod 3 (n%3). The output of this equation is then put through another modulo function where the divisor is 19 (seems random, but it is important later). Then find the natural log of this number and you have you final value.

Do this for every cell, apply some conditional formatting, and voila, you have a fractal.

Some interesting stuff:

There are three aspects of this process that can be tweaked to get different patterns.

1. Number line sequence
   • The number line can be any sequence of real numbers.
   • For the purposes of the above formula, Excel doesn't consider decimals when evaluating if a number is even or odd. 3.14 is odd, 2.718 is even.
2. Seed value
   • Seed value is the origin on the coordinate plane.
   • I like to apply recursive functions to a random seed value to generate different sequences for my number line.
3. The second Modulo Divisor
   • The second modulo divisor can be any integer greater than or equal to 19.

The first fractal in the gallery is the "simplest". It uses the positive number line from 0 to 128 and has 19 as the second modulo divisor. The rest have varying parameters which I forgot to record :(

If you take a look at the patterns I included, they all appear to have a "background". This background is where every cell begins to approximate 2.9183... Regardless of the how the above aspects are tweaked this always occurs.

This is because n=2.9183+2.9183=5.8366. Since this is an odd value (according to Excel), 3n+1 is applied (3*5.8366)+1=18.5098. If the divisor of the second modulo is >19, the output will remain 18.5098. Finally, the natural log is calculated: ln(18.5098)=2.9183. (Technically as long as the divisor of the second modulo is >(6*2.9183)+1 this holds true)

There are also some diagonal streams that are isolated from the so-called background. These are made up of a series of approximating values. In the center is 0.621... then on each side in order is 2.4304... 2.8334... 2.9041... 2.9159... 2.9179... 2.9182... and finally 2.9183... I'm really curious as to what drives this relationship.

The last fractal in the gallery is actually of a different construction. The natural log has been swapped out for Log base 11, the first modulo divisor has been changed to 7, and the second modulo divisor is now 65. A traditional number line is not used for this pattern, instead it is the Collatz Sequence of n=27 (through 128 steps) with 27 being the seed value at the origin.

n[x,y] = (x-1,y)+(x,y-1)
=IFERROR(LOG(MOD(IF(ISODD(n),(n*3)+1,MOD(n,7)),65),11),0)

This method is touchier than the first, but is just as interesting. The key part of this one is the Log base 11. The other values (seed, sequence, both modulo divisors) can be tweaked but don't always yield an "interesting" result. The background value is different too, instead of 2.9183 it is 0.6757.

What I love about this pattern is that it has a very clear "Pascality" to it. You can see the triangles! I have only found this using Log base 11.

If anyone else plays around with this I'd love to see what you come up with :)

I was looking up the Fibonacci sequence earlier today, and it seems like when it was first described, it was used for poetry in India or to estimate numbers of immortal rabbits in Europe, neither of which really seem all that useful. So it got me thinking about whether there are other discoveries that were really just interesting for centuries until someone finally discovered a practical use for them?

DOI/PMID/ISBN: https://doi.org/10.1007/s43539-022-00053-1

[URL]( https://link.springer.com/article/10.1007/s43539-022-00053-1 )

Thank you!