Important Number Sequences

fib seq on circle
Having more than a passing acquaintance with Fibonacci and the Golden Mean through my work on financial markets, I immediately looked at the sequence in a new light having thrown the Mod 9 lens over it.
For those that do not know, the Fibonacci sequence begins
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 etc.
It is the organising structure found throughout Nature and Evident In Every Living Thing and Physical Phenomena.
The interesting thing about this sequence is that by adding the previous 2 terms to get the next term in the series, the numbers adjacent approximate to a Phi / Golden Mean relationship of 1.618.
More unusually the numbers once removed are in a relationship of 2.618!
The higher up in the sequence, the closer two consecutive numbers of the sequence divided by each other will approach the golden ratio or what is called PHI, which is approximately 1: 1.618 or 0.618: 1.
The Organising Structure Found Throughout Nature. Evident In Every Living Thing.
The Golden Ratio is found throughout the human body, even in our DNA. The DNA molecule, the program for all life, measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.
I discovered to my amazement that if you go far enough out you soon realise that the Fibonacci sequence with MOD 9 applied, produces a repeating 24 number sequence:
1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1,9
I couldn’t believe it! I thought, nobody will have found that, but research unfortunately revealed that I was not the first to see this recurring pattern, but it seemed that it had only recently been discovered.
Upon further investigation I noted that if you take the second 12 numbers of the sequence and place them under the first 12 numbers we can see the beautiful symmetry in the MOD 9 sequence produced by the Fibonacci Series, where top and bottom numbers are Number Pairs.
112358437189
887641562819
Adding Totals    1000000000008
Even more interesting is when we write the Fibonacci sequence, Mod 9, like this, showing a four fold congruence in 4 columns of 6
1 1 2 3
5 8 4 3
7 1 8 9
8 8 7 6
4 1 5 6
2 8 1 9
Column 1 shows the Doubling Circuit Numbers of the 1 2 4 8 7 5
Column 2 shows 1 and 8 Number Pair.
Column 3 shows the Doubling Circuit Numbers again.
Column 4 shows the 3 6 9 Family Number Group (more about these later).
Also please note an arrangement of the 24 numbers in 8 columns of 3 clearly exhibiting the Family Number Groups when read vertically
fib 8x3
The Family of Fibonacci Sequences
Fib sequences Mod 9
Above we can see all of the Mod 9 Fibonacci Sequences and the Number Pairs are in evidence here too
1 and 8   2 and 7   3 and 6    4 and 5.
To explain if you do not yet see, let’s take a quick look at what happens if you start the Fibonacci sequence with 2 2 instead of 1 1, and again using Mod 9 for the results of the sequence, we also find a repeating 24 number pattern that exhibits symmetry.
2 2 4 6 1 7 8 6 5 2 7 9
7 7 5 3 8 2 1 3 4 7 2 9
And then arranged in 4 columns
2 2 4 6
1 7 8 6
5 2 7 9
7 7 5 3
8 2 1 3
4 7 2 9
Column 1 Doubling Circuit.
Column 2 2 and 7 Number Pair.
Column 3 Doubling Circuit.
Column 4 The 3 6 9 Family Number Group.
Now, the Fibonacci Sequence starting 3 3:
Symmetry again.
3 3 6 9 6 6 3 9 3 3 6 9
6 6 3 9 3 3 6 9 6 6 3 9
And
3 3 6 9
6 6 3 9
3 3 6 9
6 6 3 9
3 3 6 9
6 6 3 9
Interesting to see how these numbers divide themselves
Columns 1 2 and 3 show the 3 6 Number Pair
Column 4 just the 9
The 4 4 sequence: Symmetry
4 4 8 3 2 5 7 3 1 4 5 9
5 5 1 6 7 4 2 6 8 5 4 9
And
4 4 8 3
2 5 7 3
1 4 5 9
5 5 1 6
7 4 2 6
8 5 4 9
Column 1 Doubling Circuit Numbers
Column 2 4 and 5 Number Pair
Column 3 Doubling Circuit Numbers
Column 4 The 3 6 and 9 Number Group
The 5 5 Sequence: Symmetry
5 5 1 6 7 4 2 6 8 5 4 9
4 4 8 3 2 5 7 3 1 4 5 9
And
5 5 1 6
7 4 2 6
8 5 4 9
4 4 8 3
2 5 7 3
1 4 5 9
Column 1 Doubling Circuit.
Column 2 The 4 and 5 Number Pair.
Column 2 Doubling Circuit.
Column 4 The 3 6 and 9 Family Number Group.
Now it should be clear that as we progress to the higher numbered sequences they will just be mirrors of their Number Pair as we have just seen with the 4 4 and 5 5 sequence.
4 4 8 3 2 5 7 3 1 4 5 9                                                5 5 1 6 7 4 2 6 8 5 4 9
5 5 1 6 7 4 2 6 8 5 4 9                                                4 4 8 3 2 5 7 3 1 4 5 9
There is much more work to do in this area as I feel there is an enormous amount of information to be gleaned from understanding the geometry produced by all these Fibonacci sequences (and all other integer sequences) and investigating the exact proportion of the diameter of the circle where there are crossovers directly under the 9, for example.

Other Integer Sequences

I then applied the same technique to all the other important number sequences, using Mod 9, to find repeating cycles of Numbers or moduli underpinning them all.
Lucas Sequence – Mod 9
other integer sequences
24 digit recurring sequence Mod 9 with numerical symmetry.
2 1 3 4 7 2 0 2 2 4 6 1 7 8 6 5 2 7 0 7 7 5 3 8
2 1 3 4 7 2 0 2 2 4 6 1 Sum is 34
7 8 6 5 2 7 0 7 7 5 3 8 Sum is 65
9 9 9 9 9 9 9 9 9 9 9 9 Sum is 99
Background Information – Wikipedia
The Lucas Sequence is another series quite similar to the Fibonacci series that often occurs when working with the Fibonacci series. Edouard Lucas (1842-1891) (who gave the name “Fibonacci Numbers” to the series written about by Leonardo of Pisa) studied this second series of numbers: 2, 1, 3, 4, 7, 11, 18, .. called the Lucas numbers in his honour.
Pell Numbers – Mod 9
other integer sequences
24 digit recurring sequence Mod 9 with symmetry
1 2 5 3 2 7 7 3 4 2 8 9 8 7 4 6 7 2 2 6 5 7 1 9
1 2 5 3 2 7 7 3 4 2 8 9 – Sum is 53
8 7 4 6 7 2 2 6 5 7 1 9 – Sum is 64
9 9 9 9 9 9 9 9 9 9 9 9  – Sum 117
Background Information – Wikipedia
The Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2.
This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29.
The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.
Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + sq root 2.
As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.
Jacobsthal Numbers – Mod 9
other integer sequences
18 digit recurring sequence Mod 9 with symmetry
1 1 3 5 2 3 7 4 9 8 8 6 4 7 6 2 5 9
1 1 3 5 2 3 7 4 9 – Sum is 35
8 8 6 4 7 6 2 5 9 – Sum is 55
9 9 9 9 9 9 9 9 9 – Sum is 90
Background Information – Wikipedia
The Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence.
In simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that.
Padovan Sequence
other integer sequences
39 digit recurring sequence Mod 9.
1 9 9 1 9 1 1 1 2 2 3 4 5 7 9 3 7 3 1 1 4 2 5 6 7 2 4 9 6 4 6 1 1 7 2 8 9 1 8
No symmetry here but something else altogether!
Recurring Mod 9 Sequence Sum is 171
1 9 9  1 9 1 1 1 2 2 3 4 5 – Sum is 48 = 3 Mod 9
7 9 3 7 3 1 1 4 2 5 6 7 2 – Sum is 57 = 3 Mod 9
4 9 6 4 6 1 1 7 2 8 9 1 8 – Sum is 66 = 3 Mod 9
3 9 9 3 9 3 3 3 6 9 3 6 using Mod 9 addition
Instead of the customary binary symmetry we see this extraordinary type of trinary, number systemic, congruence, as we see the columns beautifully displaying the Family Number Groups so prominently. 1 4 7, 2 5 8 and 3 6 9.
This is very special.
Below is a spiral of equilateral triangles with side lengths which follow the Padovan sequence.
padovan eq triangles
Background Information
The name plastic number (het plastische getal in Dutch) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the golden ratio and silver ratio, the word plastic was not intended to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape.
This is because, according to Padovan, the characteristic ratios of the number, 3/4 and 1/7, relate to the limits of human perception in relating one physical size to another.
The plastic number, discovered by Dom Hans van der Laan (1904-91) in 1928 shortly after he had abandoned his architectural studies and become a novice monk, differs from all previous systems of architectural proportions in several fundamental ways.
Its derivation from a cubic equation (rather than a quadratic one such as that which defines the golden section) is a response to the three-dimensionality of our world.
It is truly aesthetic in the original Greek sense, i.e., its concern is not ‘beauty’ but clarity of perception.
Its basic ratios, approximately 3:4 and 1:7, are determined by the lower and upper limits of our normal ability to perceive differences of size among three-dimensional objects.
The lower limit is that at which things differ just enough to be of distinct types of size.
The upper limit is that beyond which they differ too much to relate to each other; they then belong to different orders of size.
According to Van der Laan, these limits are precisely definable.
The mutual proportion of three-dimensional things first becomes perceptible when the largest dimension of one thing equals the sum of the two smaller dimensions of the other.
This initial proportion determines in turn the limit beyond which things cease to have any perceptible mutual relation.
Proportion plays a crucial role in generating architectonic space, which comes into being through the proportional relations of the solid forms that delimit it.
Architectonic space might therefore be described as a proportion between proportions.
7 types 8 measures
The order of size embraces seven consecutive types contained between eight measures
Richard Padovan, “Dom Hans Van Der Laan and the Plastic Number”, pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.
A spiral can be formed based on connecting the corners of a set of 3 dimensional cuboids.
This is the Padovan cuboid spiral.
Successive sides of this spiral have lengths that are the Padovan sequence numbers multiplied by the square root of 2.
In mathematics the Padovan cuboid spiral is the spiral created by joining the diagonals of faces of successive cuboids added to a unit cube.
The cuboids are added sequentially so that the resulting cuboid has dimensions that are successive Padovan numbers.
The first cuboid is 1 x 1 x 1.
The second is formed by adding to this a 1 x 1 x 1 cuboid to form a 1 x 1 x 2 cuboid.
To this is added a 1 x 1 x 2 cuboid to form a 1 x 2 x 2 cuboid.
This pattern continues, forming in succession a 2x2x3 cuboid, a 2x3x4 cuboid etc.
Joining the diagonals of the exposed end of each new added cuboid creates a spiral (seen as the black line in the figure).
The points on this spiral all lie in the same plane.
The cuboids are added in a sequence that adds to the face in the positive y direction, then the positive x direction, then the positive z direction.
This is followed by cuboids added in the negative y, negative x and negative z directions. Each new cuboid added has a length and width that matches the length and width of the face being added to.
The height of the nth added cuboid is the nth Padovan number.
Connecting alternate points where the spiral bends creates a series of triangles, where each triangle has two sides that are successive Padovan numbers and that has an obtuse angle of 120 degrees between these two sides.
As the spiral unfolds, the triangles approach a form that has angles of 120 degrees, 34.61 degrees and 25.39 degrees.
padovan cubes
Tribonacci Series
tribonacci
Very interestingly the Tribonacci Series returns a 39 digit recurring sequence Mod 9 which is the same number sequence as we found in the Padovan Sequence.
9 9 1 1 2 4 7 4 6 8 9 5 4 9 9 4 4 8 7 1 7 6 5 9 2 7 9 9 7 7 5 1 4 1 6 2 9 8 1
Here we see the same effect, perhaps even more so, as the columns show the Family Number Groups
9 9 1 1  2 4 7 4 6 8 9 5 4 – Sum is 69
9 9 4 4 8 7 1 7 6 5 9 2 7 – Sum is 78
9 9 7 7 5 1 4 1 6 2 9 8 1 – Sum is 69
9 9 3 3 6 3 3 3 9 6 9 6 3 – Mod 9 Addition
Digit Sum for Mod 9 sequence is 216…. (3×72)
Background Information – Wikipedia
The Tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms.
The Tribonacci constant is the ratio toward which adjacent Tribonacci numbers tend.
It is a root of the polynomial x3 – x2 – x – 1, approximately 1.83929 and also satisfies the equation x + x?3 = 2.
It is important in the study of the snub cube.
Higher Orders
Tetranacci Numbers
The Tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are:
0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, …
The Tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomialx4 ? x3 ? x2 ? x ? 1, approximately 1.92756 and also satisfies the equation x + x?4 = 2.
Pentanacci, Hexanacci, and Heptanacci numbers have been computed.
The Pentanacci numbers are:
0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, …
Hexanacci numbers:
0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, …
Heptanacci numbers:
0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, …
Octanacci numbers:
0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, …
Nonacci numbers:
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272,…
Conclusion
It seems clear that there is information to be gleaned from a Mod 9 analysis of all integer sequences of consequence and especially the 3 dimensional aspect of both the Padovan Sequence and the Tribonacci Series ‘symmetries’.
Catalan Numbers
Talal Ghannam, in his excellent book The Mystery of Numbers points out importantly that Nature or Natural Integer Sequences all have a periodicity or modulus revealed by Mod 9  analysis.
However, when we look at Man made sequences like the Catalan sequence, which are generated by counting the different ways an n-sided polygon can be divided into triangles by connecting its vertices as shown below, no modulus is in evidence.
1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, … (Sloane’s A000108)
CatalanPolygons

Dividing Space

In the course of my research I came across these interesting sequences produced by dividing Space into Regions using Planes, Circles and Ellipses.

Mod 9 Analysis has again Proved to be Very Revealing! 

Lines Dividing a Plane
Maximal Number of Regions into which Lines Divide a Plane are given by the equation:
 1/2 ( n^2 + N + 2)
 which gives the sequence 2, 4, 7, 11, 16, 22, … (Sloane’s A000124).
dividing space
This is the same maximal number of regions into which a circle, square, etc. can be divided by lines. 
Mod 9 Analysis
line plane seq
This sequence returns a repeating a centred, 9 digit Palindrome.
1 2 4 7    2    7 4 2 1
Alert to the Trinary Congruence seen elsewhere
1 2 4
7 2 7
4 2 1
We see it here, to a degree, but this time just the 1 4 7 Family Number Group and the 2.
Notes
Term 9 – 37, 12th Prime Number and key component of the Hologram Projector of Physical Reality found in the Partition Table
Term 11 – 56 – Partitions of the Number 11 also total 56. 11 is the Number of the Earth.
Term 16 – 121 – 11^2, 11 being the Number of the Earth – 16 codons per each Amino Acid base.
Term 17  – 137 – the 33rd Prime Number  and the key number – The Fine Structure Constant 1 / 137 = 0.00 729 927
17 is the mirror of 71, the 20th Prime Number and the largest used as a factor by the Monster, largest of the Sporadic Groups.
1 4 7 Family Number Group dominates but is not exclusive
Spheres Dividing Space
Maximal Number of Regions into which Spheres Divide Space
The number of regions into which space can be divided by mutually intersecting spheres is
 1/3n (n^2 -3n +8)
giving the sequence 2, 4, 8, 16, 30, 52, 84, … (Sloane’s A046127)
spheres divide space
 
Mod 9 Analysis
spheres and space
This sequence returns a repeating 27 digit sequence !
0 2 4 8 7 3 7 3 2
6 8 1 5 4 0 4 0 8
3 5 7 2 1 6 1 6 5
No Symmetry but we can see congruence via knowledge of the Family Number Groups when the sequence is split and arranged as above. We see this trinary congruence or 3 dimensional numerical symmetry in the Padovan Sequence and the Tribonacci Sequence, which, to my mind, must therefore all be related because of this special property.
Notes
5th Term 16
6th Term 30
7th Term 52
9th Term 128 is 2^7
11th Term 260 makes me think of Time – 13 x 20 – 260 Mayan Tzolkin
17th Term 1152
27th Term 5252 at the 27th Term and 8180 at the 31st Term also resonate along this line of thinking.
Space Cube Division By Planes
The maximal number of regions into which space can be divided by n planes is defined by the equation
 1/6 (n^3+5n+6)
This give the values 2, 4, 8, 15, 26, 42, … (Sloane’s A000125), sometimes known as the “cake numbers.”
This is the same solution as for cylinder cutting.
planes and space
Mod 9 Analysis
Here we can see a 27 digit repeating sequence, Mod 9.
 1 2 4 8 6 8 6 1 3
4 5 7  2 0 2 0 4 6
7 8 1  5 3 5 3 7 0
Notes
Term 6 where the Sum Sequence = 56 which is the Sum of the two 7 based triangles (28 x 2+56) which produce a 37 Star Hexagram and a 19 Centred Magic Hexagon and the Sum Sequence – 1 = 55 The Pivot for the Hologram Projector.
Term 12 where the Sum Sequence – 1 = 792 which is the total partitions for the number 21, the mirror of 12.
Plane Division By Ellipses
Consider n intersecting ellipses. The maximal number of regions into which these divide the plane are given by the equation
2(n^2 – n + 1)
giving the sequence of 2, 6, 14, 26, 42, 62, 86, 114, … (Sloane’s A051890). 
Mod 9 Analysis
planes and ellipses
Here we can see a 9 digit recurring  sequence, Mod 9
2 2 6 5  8  6 8 5 6
No palindrome this time it would seem, but note the exclusivity of the  2 5 8 Family Number Group here with the 6 and the Trinary congruence again
2 2 6
5 8 6
8 5 6
I noted that all the numbers in this sequence appear to be multiples of prime numbers, either 2 x or 6 x various Prime Numbers in the sequence
2 x 3 = 6
2 x 7 = 14
2 x 13 = 26
6 x 7 = 42  – NB Also 3 x 14 and 2 x 21
2 x 31 = 62
2 x 43 = 86
6 x 19 = 114 NB Also 2 x 57 and 3 x 38
2 x 73 = 146 (73 is 21st Prime)
26 x 7 = 182 NB Also 13 x 14
6 x 37 = 222 (37 is 12th Prime)
2 x 133 = 266 etc

Platonic Solid Numbers

Platonic Numbers

Tetrahedral Numbers
1 4 1 2 8 2 3 3 3 4 7 4 5 2 5 6 6 6 7 1 7 8 5 8 0 0 0
1 4 1 2 8 2 3 3 3
4 7 4 5 2 5 6 6 6
7 1 7 8 5 8 0 0 0
Octahedral Numbers
1 6 1 8 4 2 6 2 3 4 0 4 2 7 5 0 5 6 7 3 7 5 1 8 3 8 0
1 6 1 8 4 2 6 2 3
4 0 4 2 7 5 0 5 6
7 3 7 5 1 8 3 8 0
Cube Numbers
1 8 0
Icosahedral Numbers
1 3 3 7 3 6 4 3 0
1 3 3
7 3 6
4 3 0
Dodecahedral Numbers
1 2 3 4 5 6 7 8 0
1 2 3
4 5 6
7 8 0

Centred Platonic Numbers

Platonic Centred Numbers
Centred Tetrahedral
1 5 6 8 6 4 6 7 2 4 8 0 2 0 7 0 1 5 7 2 3 5 3 1 3 4 8
1 5 6 8 6 4 6 7 2
4 8 0 2 0 7 0 1 5
7 2 3 5 3 1 3 4 8
Centred Octahedral
1 7 7 0 3 6 8 8 5 7 4 4 6 0 3 5 5 2 4 1 1 3 6 0 2 2 8
1 7 7 0 3 6 8 8 5
7 4 4 6 0 3 5 5 2
4 1 1 3 6 0 2 2 8
Centred Cube
1 0 8
Centred Icosahedral Numbers
1 4 1 3 3 3 5 2 5 7 1 7 0 0 0 2 8 2 4 7 4 6 6 6 8 5 8
1 4 1 3 3 3 5 2 5
7 1 7 0 0 0 2 8 2
4 7 4 6 6 6 8 5 8
Centred Dodecahedral Numbers
1 6 2 4 0 5 7 3 8
1 6 2
4 0 5
7 3 8

 

Next – Prime & Composite Numbers

 

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