Experimental Mathematics: Finding Number Patterns

[Natus Videre]

One day, you sit down at your desk, you grab a piece of paper and a pencil, and you start writing down the natural numbers.

1,2,3,4,5,6...

They are very familiar to you. You have known them for a long time.
But now you wonder if there is more to them than meets the eye...

You start adding them, multiplying them, concatenating them, reordering them... anything which comes to your mind.
Your paper becomes crowded with numbers. You are overwhelmed by what you find: very small numbers, very big numbers, even numbers, odd numbers, prime numbers, repeating digits, exponential growth, factorials, geometric continuations, different bases, numbers sharing unique properties, square roots, digital roots, and continued fractions.

After some contemplation, you pick a number sequence near the edge of paper.

1,3,6,10,15,21,...

But you can't quite recall how you found it. So you perform a few operations, but nothing comes close to the sequence.
Thankfully, you are aware of The On-Line Encyclopedia of Integer Sequences (OEIS) and so you type the mysterious sequence into the search bar:

1,3,6,10,15,21 - OEIS

Triangular numbers, how sweet!

Suddenly, you remember how you generated the sequence from 1,2,3,4,5,6:

1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21

Fantastic, is there a geometrical pattern lurking there?



Triangles, triangles everywhere!
You wonder if there is more to it than meets the eye. You apply the same procedure starting from 1,3,6,10,15,21 this time.

1 = 1
1 + 3 = 4
1 + 3 + 6 = 10
1 + 3 + 6 + 10 = 20
1 + 3 + 6 + 10 + 15 = 35
1 + 3 + 6 + 10 + 15 + 21 = 56

You obtain 1,4,10,20,35,56,... the Tetrahedral numbers!
But you continue, you are relentless!

1 = 1
1 + 4 = 5
1 + 4 + 10 = 15
1 + 4 + 10 + 20 = 35
1 + 4 + 10 + 20 + 35 = 70
1 + 4 + 10 + 20 + 35 + 56 = 126

You discover the Pentatope numbers: 1,5,15,35,70,126,...
Hmm.. but how do these sequences fit together?



Pascal's triangle—we meet again!
...and thus we have another starting point, but this time we won't forget where we started from.

 

[mkrnhr]

Pascal's triangle is remarkable indeed
If one sums the diagonals like this (pardon my artistic skills)

we recover our old friend Fibonacci. It can be demonstrated why it is so but it's nice to see.

 

[mkrnhr]

Another property of the Pascal triangle I came across once: If even numbers are removed and odd numbers retained, one obtains the Sierpinski fractal:

 

[Natus Videre]

we recover our old friend Fibonacci. It can be demonstrated why it is so but it's nice to see

Another property of the Pascal triangle I came across once: If even numbers are removed and odd numbers retained, one obtains the Sierpinski fractal:

Yes, it's fascinating!

We have derived most of Pascal's triangle from natural numbers. Is there a way to derive natural numbers in a similar way?
The answer is yes!

1 = 1
1 + 1 = 2
1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
1 + 1 + 1 + 1 + 1 = 5
1 + 1 + 1 + 1 + 1 + 1 = 6

Pascal's triangle can be entirely derived from the number 1. Now is there a link between 1 and the Fibonacci sequence (1,1,2,3,5,8,13)?

Using continued fractions, it is possible to calculate the golden ratio (1.618...) just with the number 1!


Did you know that you can take any two numbers, apply the 'Fibonacci' algorithm (sum of two last numbers), and approach the golden ratio?

For example, take 5 and 17.
5 + 17 = 22
17 + 22 = 39
22 + 39 = 61
39 + 61 = 100
61 + 100 = 161
100 + 161 = 261
161 + 261 = 422

Now, check the ratio between successive numbers:
39/22 = 1.7727...
61/39 = 1.5641...
100/61 = 1.6393...
161/100 = 1.61
261/161 = 1.6211..
422/261 = 1.6168... (already getting very close to 1.618..)

Hmm... successive operations with two numbers... the number 2 keeps resurfacing...

Powers of 2


Didn't the C's mention that 'squaring' represents 'balance?'

Q: Okay, one interesting thing that we just discovered was that Hyakatuke and Hale Bopp both crossed the eye of Medusa, the star Algol, on April 11th exactly one year apart. What is the significance of this?
A: You must remember mosaic, matrix... When you are on the verge of quantum changes or discovery, the realities begin to reveal their perfectly squared nature to you.

Q: Is that the only thing you want to remark about the crossing of the comets in front of the eye of Medusa?
A: Can you not picture all reality as a curving and bobbing journey through a transparent, undulating matrix mosaic?

Q: Well, do you have anything else to say about Andromeda? (It's VERY HOT in here!) Okay, Medusa 11. So, this was 11 of the 33, and assuming that you were not saying that there were 11 heads, but that Medusa was one of three heads, is that what we are getting at here, that there are three heads and Medusa was one?
A: Or both times 2.

Q: What do you mean? I don't understand.
A: Both times 2 is your square, my dear. In other words, perfect balance.

But there is also 11 (eleven or ... one-one, the 'balanced' concatenation?). Notice the symmetry along the vertical axis of the triangle, as the 1's 'wrap' the rest of the numbers. One left part, one right part... 11. Dual meaning.

Powers of 11

Pascal's Triangle

To build the triangle, start with 1 at the top, then continue placing numbers below it in a triangular pattern. Each number is the numbers directly above it added together.

www.mathsisfun.com

 

[mkrnhr]

All sequences of the form F_(n+1) = F_(n) + F_(n-1) will have F_(n+1) / F_(n) tend to the golden ratio at large n. Fibonacci starts with 1,1, Lucas with 2,1, etc.
There are also other "algorithms" that converge to other special numbers as well.
The interesting thing with Fibonacci is that it appears in the infinite fraction of the golden ratio, the infinite fraction is apparently the simplest (just ones) and yet it produces the most irrational number, i.e. the most "difficult" to approximate by a simple ratio. Not sure what it means, if it means anything, but it's fascinating.

 

[John G]

Couple of my favorite Pascal Triangle patterns:

The rows with the second number being a prime number have all the non-one numbers in the row being a multiple of that prime as in the row with 5 as the second number has a 10 in the row and the row with 7 as the second number has 21 and 35 in the row.

The rows with a middle number like the one with 70 and the one with 20 come from a sum of squares as in
70=1+16+36+16+1= 1^2 + 4^2 +6^2 + 4^2 + 1^2 with the 1 4 6 4 1 being the row where the 4 is half of the 8 from the row with 70 in the middle. For the row with 20 it's 20=1^2 + 3^2 +3^2 +1^2 where the 3 is half of the 6 from the row with 20 in the middle (probably a better way to say this).

 

[heinrich]

Couple of my favorite Pascal Triangle patterns:

The rows with the second number being a prime number have all the non-one numbers in the row being a multiple of that prime as in the row with 5 as the second number has a 10 in the row and the row with 7 as the second number has 21 and 35 in the row.

The rows with a middle number like the one with 70 and the one with 20 come from a sum of squares as in
70=1+16+36+16+1= 1^2 + 4^2 +6^2 + 4^2 + 1^2 with the 1 4 6 4 1 being the row where the 4 is half of the 8 from the row with 70 in the middle. For the row with 20 it's 20=1^2 + 3^2 +3^2 +1^2 where the 3 is half of the 6 from the row with 20 in the middle (probably a better way to say this).

in this context i recall the site i have mentioned earlier which is "bibleetnombres" and which can be found in this forum. there also, many numbers are correlated with words and interesting sequences are provided.

 

[Natus Videre]

From The remarkable number 1/89:

The Fibonacci series

Assume that a pair of rabbits will breed another pair of rabbits every month of their lives, after sexual maturity. Assume also that rabbits become sexually mature after two months. Starting with one newborn pair of rabbits, describe how many rabbits you get in successive months.

At the end of the first month you have 1 pair. After another month, you still have 1 pair. However, at the end of the third, the first pair has become sexually mature, and produced offspring, so you have 2 pair. At the end of the next month, the first pair reproduces again, while the second pair remains immature, so you have a total of 3 pair.

Proceeding in this fashion, one sees that the following list of numbers describes the breeding of the rabbits:

1,1,2,3,5,8,13,21,34,55,89,144,...

Here is a diagram I made to visualize the Fibonacci breeding of the rabbits (1,1,2,3,5,...). Each white circle represents an immature pair of rabbits. Each black circle represents a mature pair of rabbits. To make it easier to count the total population, I inscribed a star in every circle that is relevant to the calculation (once a white circle transforms (grows) into a black (mature) circle, it is no longer counted towards the total).


Notice how there are two phenomena at play: the maturation (continuation) process and the breeding (splitting) process. Still, there is a seed from which everything grows. The pattern has cumulative, recursive, and exponential properties!

Here is another perspective.

For example, consider the second and third rows from the bottom, the number 8 appears twice on the left branch (continuation), while the number 13 is split into 5 and 8 on the right branch (breeding).

Did you know that 1/89 encapsulates the Fibonacci sequence in its decimal representation?

The Punch Line​

The decimal expansion of 1/89 is just the Fibonacci series, added together in an appropriate fashion.

Specifically, think of the Fibonacci series as being a sequence of decimal fractions, arranged so the right most digit of the nth Fibonacci number is in the n+1th decimal place. Then add:

.01
.001
.0002
.00003
.000005
.0000008
.00000013
.000000021
.0000000034
.00000000055
.000000000089
.0000000000144
+
----------------
.01123595505...

As you can easily check, 1/89 = .01123595595... Bizarre, eh?

The number 89 is both a Fibonacci number and a prime number!

1,1,2,3,5,8,13,21,34,55,89,144,...
1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,...

Stay tuned for more surprises!

 

[Natus Videre]

The 'Fibonacci' algorithm sums two preceding numbers to obtain the next one.
What if we summed the three preceding numbers instead?

We would get the Tribonacci numbers!

...which are also part of the N-acci family (starting numbers in bold):

Fibonacci: 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,...
Tribonacci: 0,0,1,1,2,4,7,13,24,44,81,149,274,504,...
Tetranacci: 0,0,0,1,1,2,4,8,15,29,56,108,208,401,...
Pentanacci: 0,0,0,0,1,1,2,4,8,16,31,61,120, 236,464,...
Hexanacci: 0,0,0,0,0,1,1,2,4,8,16,32,63,125,248,492,...
Heptanacci: 0,0,0,0,0,0,1,1,2,4,8,16,32,64,127,253,504,...
Octonacci: 0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,255,509,...
Enneanacci: 0,0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,256,511,...
and so on...

Do you see any patterns?

Spoiler: Pattern

The powers of 2 make an appearance again!

1,2,4,8,16,32,64,128,256,...

The more "N" in "N-acci" increases, the closer we get to the famous "doubling" sequence!

 

[Musicinventor]

The book; The Mathematical Basis of The Arts by Joseph Schillinger, discusses multiple number series and their application to the arts. It is well worth the read and available as a free pdf download from multiple sites. The book was published in 1949 and I don’t know of it being republished over the years. I do have a pdf but it’s 46mb and probably to large to download. If there is someplace on the forum where it can be accessible to other members please let me know. As of now, I’m unsure of how to copy and post individual, pertinent, pages.

 

[Sirius]

https://twitter.com/x/status/1711262717560971405

 

[Musicinventor]

Do you see any patterns?

From Fibonacci on done I see the pattern of numbers added going 2 3 4 5 6 7 8 9 osit…

 

[Natus Videre]

The book; The Mathematical Basis of The Arts by Joseph Schillinger, discusses multiple number series and their application to the arts. It is well worth the read and available as a free pdf download from multiple sites. The book was published in 1949 and I don’t know of it being republished over the years. I do have a pdf but it’s 46mb and probably to large to download. If there is someplace on the forum where it can be accessible to other members please let me know. As of now, I’m unsure of how to copy and post individual, pertinent, pages.

Wow, very interesting book! I managed to find it.
Could you post the page numbers that you find relevant to the discussion?

 

[Musicinventor]

Could you post the page numbers that you find relevant to the discussion?

Chapter 7 page 38, and chapter 2 page 85, seem to me the most relevant. As the book discusses mathematical aspects of all the arts the entire book strikes me as being a form of experimental mathematics. I have two sets, one original publication and the second recently republished, of the hardback two volume set, The Schillinger System of Musical Composition, which I use extensively when I’m composing. I find that every piece is an experiment, both in the musical and mathematical sense. I’ve attached a few examples of my use of the Fibonacci series. I would like to include other examples but it would take a month in Sundays to find those notes!

 

[Natus Videre]

Chapter 7 page 38, and chapter 2 page 85, seem to me the most relevant. As the book discusses mathematical aspects of all the arts the entire book strikes me as being a form of experimental mathematics.

It's intriguing how there seems to be an omnipresent interplay between continuity and discontinuity. We separate numbers by commas, we concatenate them, we order them as we please, but somehow, no matter how hard we try to isolate them, they remain harmoniously linked to each other.

Can a sequence describe everything there is to know about numbers?

Many, many patterns spring from the inevitable number 1...