Fibonacci & Other Integer Sequences – Mod 9
by Anthony Morris
Please see the accompanying spreadsheet
1.618 - The Golden Mean - PHI: The Organising Structure Found Throughout Nature.
Evident In Every Living Thing.
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.
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.
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!
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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 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 then 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 where top and bottom numbers are number pairs as described in the Introductory paper - http://archive.org/details/IntroductionToNoCoincidence.
112358437189
887641562819
Adding Totals 1000000000008.
Even more interesting is when we write the Fibonacci sequence, Mod 9, like this:
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).
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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 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
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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
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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, so I won’t bore you with any more.
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 of the above and investigating the exact proportion of the diameter where there are crossovers directly under the 9, for example.
Anthony Morris – Market Harmony Ltd – [email protected]
Dec 2013 Update
A common mistake is thinking that the ratios of consecutive Fibonacci numbers are the only numbers that can create the Golden Mean. The fact is, ANY sequence of numbers, starting with ANY random two numbers, you will still eventually get to the Phi ratio if you divide the latter into the former. The same is true even if you begin with a negative number.
-8 4 -4 0 -4 -4 -8 -12 -20 -32 -52 -84 -136 by now we see that 136 / 84 = 1.619047
The golden mean ratio ALWAYS reasserts itself to control the numbers - it is inherent to any sequence of numbers formed as an addition of previous two numbers after the 9th number in the sequence.
However the 1 1 2 3 5 8 13 is the natural unfolding in the universe and is what makes the Fibonacci numbers in this sequence so special.
In wondering about this sequence I wondered what terms could come before the zero and leave the positive side of the equation unaffected and saw a rather nice sequence – there is a type of symmetry being displayed here that doesn’t seem very normal because of the alternating positive and negative value numbers as we go beyond 0
-21 13 -8 5 -3 +2 -1 1 0 1 1 2 3 5 8 13 21
The above model contains 17 Fibonacci terms 2 full octaves of eight terms either side of their fulcrum point of balance 0, beginning with -21 and 13 ending with 21. I feel this is pretty deep and essential….
To the left of zero, the total of the terms is -12 and to the right the total is 54
Addition of these terms gives you a total of 42 (2x21) which is rather elegant I would say.
So just what did Douglas Adams know?
Note
Mod 9 -3 4 -8 5 -3 +2 -1 1 0 1 1 2 3 5 8 4 3
Total is 24
21 is the key number in my paper Partition Theory & The Hologram Projector of Physical Reality.
Anthony Morris – Market Harmony Ltd – [email protected]
Inter Fibonacci Number Effects
What follows is a new look at the Fibonacci Sequence and I believe I have turned up some interesting info here using similar simple techniques as before to be able to understand what we are actually looking at.
We all know that this sequence approximates to the Golden Mean or Phi but actually I think it is important to examine exactly the numbers as they relate to each other and all other numbers.
The best way to do this seems to be by division.
Number 1
The first 2 terms of the Fibonacci Sequence, the higher divided by the lower:
1/0 = Infinity Reverse 0/1 = Infinity
Number 2
The next two in the sequence, the higher divided by the lower number
1/1 = 1 Reverse 1/1 = 1
Number 3
2/1 = 2 Reverse 1/2 = 0.5
Number 4
3/2 = 1.5 Reverse 2/3 = 0.666 R
Number 5
5/3 = 1.6666 R Reverse 3/5 = 0.6
Number 6
8/5 = 1.6 Reverse 5/8 = 0.625
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Number 7
13/8 = 1.625
Reverse
8/13 = 0.615384 R
615
384
999
Symmetry
Digit Sum of recurring decimal = 27
Number Pairs
0 9 x 0
1 8 x 1
2 7 x 0
3 6 x 1
4 5 x 1
Number 8
21/13
= 1.615384 R
615
384
999
Symmetry
Digit Sum of recurring decimal = 27
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Number Pairs
0 9 x 0
1 8 x 1
2 7 x 0
3 6 x 1
4 5 x 1
Reverse
13/21 = 0.619047 R
Number Pairs
6 1 9
0 4 7
But No Symmetry!!
A 6 digit recurring decimal without symmetry is a unique attribute of the number 21 and its multiples.
21, the product of 3 and 7 is a very special number as I have discovered in the Partition Table paper.
Digit Sum of recurring decimal = 27
Number 9
34/21 = 1.619047 R
Number Pairs
6 1 9
0 4 7
No Symmetry, as above
Digit Sum of recurring decimal = 27
Reverse
21/34 = 0.61 7647058823529411 R
76470588 Sum is 45
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23529411 Sum is 27
99999999
Digit Sum of recurring decimal = 72
Symmetry
16 Digit Recurring decimal which you would expect for 1/17.
34 is of course a multiple of 17
Number Pairs
0 9 x 1
1 8 x 2
2 7 x 2
3 6 x 1
4 5 x 2
NB Number Pairs are in the same frequency as 1/17.The recurring decimal is exactly the same with the first 4 digits moved to the end….. 1/17 = 0.0588235294117647 R
SYMMETRY
05882352 – Sum is 33 94117647 – Sum is 39 99999999
Note the different Sums of top and bottom. Digit Sum is 72
Number Pairs
1 x 0 9 pair 1 x 3 6 pair 2 x 1 8 pair 2 x 2 7 pair 2 x 4 5 pair
Here we can see a nicely balanced measure of ‘Pure Prime’ Number Pairs 1 8, 2 7 and 4 5 with a less but equally tempered 3 6 and 0 9.
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Number 10
55/34 = 1.61 7647058823529411 R
76470588 Sum is 45
23529411 Sum is 27
99999999
Digit Sum of recurring decimal = 72
Symmetry
Number 11
89/55 = 1.6 1818 R
Reversed
55/89 = 0.61797752808988764044943820224719
6179775280898876404494 = 123 = 6 Mod 9
3820224719101123595505 = 75 = 3 Mod 9
9999999999999999999999
44 digit recurring decimal with symmetry
Digit total is 198
Number Pairs
09 x 5
36 x 2
18 x 5
27 x 5
45 x 5
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Number 12
144/89 = 1.6179775280898876404494382022471910112359550561
As above
Reversed
89/144 = 0.618 0 5555 R
Such an interesting number.
The phi number of 0.618 wholly presented separated by the axis of the 0 9 and recurring on Phive, the first Prime number in physical reality.
144 being a multiple of 72 will produce single digit recurring decimals.
Number 13
233/144 = 1.618 0 55555 R
Similarly here, the Phi number is wholly presented.
Nothing much of particular interest after the 13th Term but it needs checking to confirm that it’s all about these first 13 terms.
Reverse
144/233 = 0.618 0 2575107296137339055793991416
0.618 02575107296137339055793991416309012875536480686695278969957081545064377682 403433476394849785407725321888412017167381 – Sum is 545
97424892703862660944206008583690987124463519313304721030042918454935622317 596566523605150214592274678111587982832618 – Sum is 499
Symmetry in the 232 digit recurring decimal.
Number 14
377/233 = 1.618 0 257510729613733905579399142
Number 15
610/377 = 1.618 0 371352785145888594164456233
Number 16
987/610 = 1.618 0 327868852459016393442622951
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Other Integer Sequences Showing Repeating Sequences Mod 9
Lucas Mod 9 Pell Mod 9 Jacobsthal Mod 9 Padovan Mod 9 2 2 1 1 1 1 1 1 1 1 2 2 1 1 0 9 3 3 5 5 3 3 0 9 4 4 12 3 5 5 1 1 7 7 29 2 11 2 0 9 11 2 70 7 21 3 1 1 18 0 169 7 43 7 1 1 29 2 408 3 85 4 1 1 47 2 985 4 171 9 2 2 76 4 2378 2 341 8 2 2 123 6 5741 8 683 8 3 3 199 1 13860 9 1365 6 4 4 322 7 33461 8 2731 4 5 5 521 8 80782 7 5461 7 7 7 843 6 195025 4 10923 6 9 9 1364 5 470832 6 21845 2 12 3 2207 2 1136689 7 43691 5 16 7 3571 7 2744210 2 87381 9 21 3 5778 0 6625109 2 174763 1 28 1 9349 7 15994428 6 349525 1 37 1 15127 7 38613965 5 699051 3 49 4 24476 5 93222358 7 1398101 5 65 2 39603 3 225058681 1 2796203 2 86 5 64079 8 543339720 9 5592405 3 114 6 103682 2 1311738121 1 11184811 7 151 7 167761 1 3166815962 2 22369621 4 200 2 271443 3 7645370045 5 44739243 9 265 4 439204 4 18457556052 3 89478485 8 351 9 710647 7 44560482149 2 178956971 8 465 6 1149851 2 357913941 6 616 4
1860498 0 715827883 4 816 6
3010349 2 1081 1
4870847 2 1432 1
7881196 4 1897 7
12752043 6 2513 2
20633239 1 3329 8
4410 9
5842 1 7739 8
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Lucas Sequence
24 digit recurring sequence Mod 9 with symmetry
Recurring Mod 9 Sequence Sum is 99
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
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
24 digit recurring sequence Mod 9 with symmetry
Recurring Mod 9 Sequence Sum is 117
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
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 + √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.
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Jacobsthal Numbers
18 digit recurring sequence Mod 9 with symmetry
Recurring Mod 9 Sequence Sum is 90
1 1 3 5 2 3 7 4 9 – Sum is 35
8 8 6 4 7 6 2 5 9 – Sum is 55
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
39 digit recurring sequence Mod 9.
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
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
Instead of a 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.
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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 curcial 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.
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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 1x1x1.
The second is formed by adding to this a 1x1x1 cuboid to form a 1x1x2 cuboid.
To this is added a 1x1x2 cuboid to form a 1x2x2 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.
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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.
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Tribonacci Series
Term Tribonacci Mod 9 1 0 9 2 0 9 3 1 1 4 1 1 5 2 2 6 4 4 7 7 7 8 13 4 9 24 6 10 44 8 11 81 9 12 149 5 13 274 4 14 504 9 15 927 9 16 1705 4 17 3136 4 18 5768 8 19 10609 7 20 19513 1 21 35890 7 22 66012 6 23 121415 5 24 223317 9 25 410744 2 26 755476 7 27 1389537 9 28 2555757 9 29 4700770 7 30 8646064 7 31 15902591 5 32 29249425 1 33 53798080 4 34 98950096 1 35 181997601 6 36 334745777 2 37 615693474 9 38 1132436852 8 39 2082876103 1 40 3831006429 9 41 7046319384 9
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42 12960201916 1 43 23837527729 1 44 43844049029 2 45 80641778674 4 46 148323355432 7 47 272809183135 4 48 501774317241 6 49 922906855808 8 50 1697490356184 9 51 3122171529233 5
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.
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
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
Digit Sum for Mod 9 sequence is 216.... (3x72)
Tribonacci Sum Sequence also has a 39 digit recurring sequence Mod 9
9 9 1 2 4 8 6 1 7 6 6 2 6 - Sum is 67
6 6 1 5 4 2 3 1 7 3 3 5 3 - Sum is 49
3 3 1 8 4 5 9 1 7 9 9 8 9 - Sum is 76
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.
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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 polynomial x4 − 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,...
The limit of the ratio of successive terms of an n-nacci series tends to a root of the equation
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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’.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Anthony Morris – Market Harmony Ltd – [email protected]