Fibanacci * x mod m

found discussion on 3-6-9 on web, if x3 to fibonacci sequence and reduce to single digit or mod 9
the result is octave of "3-3-6-9-6-6-3-9" or "3-3-6-0-6-6-3-0" by modulo method which 0=9, no remainder.
I try to vary multiplier (x) from 2-24 amd m = 9, 11, 12
The pattern of numbers appearance frequency are symetry and beautiful.

Fibonacci Mod 12 R2

Afetr consider pisano period on pi(m)=24, mod 9 have number 1 & 8 apprear 5 time
also mod 12 , 1 & 5 appesr 5 time. This should be beautiful pattern hidden in mod 12 as well.
Today I found such pattern...

Fibonacci Pisano Pattern

from pisano period up to m=24.
for m=2,4,11,19,22    there are no zero in the middle of cycle.
there are reverse sequences of 1,2 ,3.......as mark by bold block
there is clearly trand of some data.

Fibonacci Mod 12

After rework on pisano mod n up to 24, I try to draw N-gram mod 11 but not found beautiful pattern.
I relook to mod 12, it has 5 symetrical patterns by using the same method of mod 9 enneagram

Pythagoras Triplet

From a^2 + b^2 = c^2
If a = prime, then c = b+1, or p^2 + b^2 = (b+1)^2
If a not prime, b & c shall be prime except: 3 set of triplet (for first 100 number)
i.e. 56, 33, 65
      63, 16, 65
      77, 36, 85

The plot of p & a is highly correlation.
But the plot of a & b for non-prime quite scatter.

Fibonacci Rabbit in The Grass Field

A = B (MOD C)
A = 1,2,3,..........
B = +/-1
C = 2 to 9

The red pattern is B which continuously alaign as straigth line on excell work sheet
The green pattern is B which diagonal contact of each cell.

It look like grass field and have cycle of 2520.
In the cycle have internal mirror image.
The space between red line some of them are fibonacci number:
55, 233, 55, 1, 34, 55, 1, 34, 55, 1....1, 55, 34, 1, 55, 34, 1, 55, 233, 55
As I shown as 7 rabbit on left side and 7 ones on the right side.

Pythagoras Pattern

During hunting for Pythagoras sequence, I found beautiful pattern.
It is the plot of integer deviation vs N 1 to 10000, when the deviation is zero, then the integer number
that satisfy pythagoras equation is founded.

Fibonacci & DNA

Frequently Fibonacci appears as spirals and mirror images
I found relation from web:

Enneagram & Earth

Fibonacci sequence tilt 22.5 degree to generate enneagram.
Earth tilt 23.44 degree to generate life.

Fibonacci & Resistor Network 7

For N=5, R=1,2,3,4,5
It has one possible circuit i.e.bridge circuit.
I found info in blogspot to convert circuit from delta to star (or wye)
But it look like N=6
I found 10 possible resistance values

Fibonacci Mod m

From Pisano periods, the well known Enneagram generated from Fibonacci mod 9.
I try to find out for enneagrams base on mod m.
For mod 6 & mod 12 the pi(m)=24 same as mod 9.
For m=6, no interesting patterns apprear.
For m=12 , 1-8-7-5-4-11 create symetrical pattern. It should have hidden beautiful pattern.

Fibonacci Pisano Periods

From fibonacci pages of mathworld, there are pisano period , pi(m)
I use excell to calculate remainder from divider m=2 to 12
At m=9, the sequence is same as Fibonacci sequence that generate Enneagram numbers.
Also each set of cyclic remainder start with "1, 1" and end with "0"
They have zero at center show in blue color, except m=2 & m=4

Fibonacci & Resistor Network 6

I think sloane's sequences of resistor network did not include circuit #5.
Which have two type circuit # 5/1 & 5/2

N    A005840  A051045
1            1               1
2            2               2
3            8               8
4           46              44
4           52              48  (Add 6 distinct circuit and 4 Possible Resistance values)

Why they excuded circuit no.5 ?
Circuit 5/1 can be modified from circuit 3 or 8
      and circuit 5/2 modified from circuit 2 or 7

Fibonacci & Resistor Network 5

Use Excell to confirm all possible values.
Confirm 47 values, or 43 values excluded 1,2,3,4
Why Sloane's A051045 showed 44 for n=4?
If n=5, it has 298 possible values....try later....

Fibonacci & Resistor Network 4

Afetr try to solve Resistor net work N=4, R=1,2,3,4
Found 47 possible R values, or 43 values exclude 1,2,3,4
But the solution shall be 44 as Sloane's A051045.
I will try to re-check later.

Fibonacci & Resistor Network 3

The Resistor circuit for N=3, R=1,2,3 ohm:

For N=4, R=1,2,3,4 ohm ; the possible value shall be 44 as Sloane's A051045
I still don't clear why N=4 , the distinct arrangement is 46 (A005840)
I will try to dig out later.

Fibonacci & Resistor Network 2

The last table of Resistor network topic of "mathworld" made me in doubt on R=3 & R=4
As the table give only 4 possible value for N=3, but the sequence is 8.
I try to dig out and found they are actually have all possible valve = 8 i.e.

6/11, 5/6, 4/3, 3/2, 11/5, 11/4, 11/3, 6

(underline additional values)

Fibonacci & Resistor Network

Found Fibonacci relation with Resistor network from mathworld.
At n=4, I think "1 ohm" should be in the possible resistance list as it is one of combination of 4 '1ohm' resistors.
Therefore n=4,5.. will have 9, 18.. possible resistances.

 fibonacci number 1,2,3,5.....

n=3 , 4 possible value

n=4, 9 possible value

n=6, 18 possible value

             

Pythagoras sequence 5

For triangle leg ratio 1:1

observe that the 2nd-round start at Fibonacci number "13"

Pythagoras sequence 4

For golden triangle leg ratio 2:1

observe strat 2nd-round at Fibonacci number "21"

Pythagoras sequence 3

Try to sketch as found picture on google:

very beautiful spiral
observe the 2nd-round start near Fibomacci number "34"

Fibonacci & Pythagoras

I found relation form "mathworld.wolfram.com"

Pythagoras sequence 2

start from triangle 3-4-5, 5-12-13...
20-21-29 rotate 180degree from original 3-4-5

Fibonacci sequence & Enneagram generation 10

What are two set of number which create minimum void (non 1 2 3 ..order)?
When arrange as table , I would like to call MATRIX
Are they 142857 & 693396 ?

Fibonacci sequence & Enneagram generation 1

124875 has more details at website of Marko Rodin
He apply as coil which is higher magnetic field more than normal one.
It may be applied for fusion reactor...

Fibonacci sequence & Enneagram generation 9

The CC' appear only 1 time

I think this should be clear why enneagram link to personality.
Because Fibonacci link to life.

Fibonacci sequence & Enneagram generation 8

Some pattern can be observed.
But it quite complex as fibonacci is basic number for life. Life is complex.
Is this link to other thing e.g. DNA...?

Fibonacci sequence & Enneagram generation 7

To simplify, split them as 3 digit group:

Therefore 142857 = AA'

Fibonacci sequence & Enneagram generation 6

Fibonacci sequence & Enneagram generation 5

Fibonacci sequence & Enneagram generation 4

Fibonacci sequence & Enneagram generation 3

There are another two type i.e. type 2 & 3 generate by rotation the start point.
Total 24 set of number generated which included type 2 & 3.