Fibonacci 24 Repeating Pattern

The Fibonacci sequence has a pattern that repeats every 24 numbers.

Numeric reduction is a technique used in analysis of numbers in which all the digits of a number are added together until only one digit remains.  As an example, the numeric reduction of 256 is 4 because 2+5+6=13 and 1+3=4.

Applying numeric reduction to the Fibonacci series produces an infinite series of 24 repeating digits:

1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9

If you take the first 12 digits and add them to the second twelve digits and apply numeric reduction to the result, you find that they all have a value of 9.

 

1st 12 numbers112358437189
2nd 12 numbers887641562819
Numeric reduction – Add rows 1 and 29999999999918
Final numeric reduction – Add digits of result999999999999

This pattern was contributed both by Joseph Turbeville and then again by a mathematician by the name of Jain.

We would expect a pattern to exist in the Fibonacci series since each number in the series encodes the sum of the previous two.  What’s not quite so obvious is why this pattern should repeat every 24 numbers or why the first and last half of the series should all add to 9.

For those of you from the “Show Me” state, this pattern of 24 digits is demonstrated in the numeric reduction of the first 73 numbers of the Fibonacci series, as shown below:

Fibonacci Number

Numeric reduction by adding digits
1st Level2nd LevelFinal Level
Example: 2,5842+5+8+4=191+9=101+0=1
0000
1111
1111
2222
3333
5555
8888
13444
21333
34777
551011
891788
144999
233888
3771788
610777
9872466
1,5972244
2,58419101
4,1811455
6,7652466
10,9462022
17,7111788
28,65728101
46,3682799
75,02519101
121,39319101
196,41829112
317,8112133
514,2292355
832,0401788
1,346,2693144
2,178,3093033
3,524,5783477
5,702,88737101
9,227,4653588
14,930,3522799
24,157,8173588
39,088,1694488
63,245,9864377
102,334,1552466
165,580,1413144
267,914,29646101
433,494,4374155
701,408,7333366
1,134,903,17029112
1,836,311,9033588
2,971,215,07337101
4,807,526,9765499
7,778,742,04955101
12,586,269,02546101
20,365,011,07429112
32,951,280,09948123
53,316,291,1734155
86,267,571,2725388
139,583,862,44558134
225,851,433,71748123
365,435,296,1625277
591,286,729,87973101
956,722,026,0414488
1,548,008,755,9205499
2,504,730,781,9615388
4,052,739,537,8816288
6,557,470,319,8426177
10,610,209,857,7235166
17,167,680,177,56567134
27,777,890,035,28873101
44,945,570,212,85359145
72,723,460,248,1415166
117,669,030,460,99465112
190,392,490,709,1356288
308,061,521,170,12946101
498,454,011,879,2647299

 

Thanks to Joseph Turbeville for sending “A Glimmer of Light from the Eye of a Giant” and to Helga Hertsig for bringing Jain’s discovery of this pattern to my attention.

Comments

  1. tom barnett says

    Hi,

    regarding this Repeating pattern in the Fibonacci Series.
    I have taken analysis of this a few stages further if you would care to take a look.
    The ‘adding up to 9′ thing is just one possible pattern to discover. But there are many more to be found.
    You can download some of my analysis here:

    http://vbm369.ning.com/forum/topics/fibonexus-and-lucanexus-continued

    scroll down and download the files. You might find the
    “fib divided into fib mod9 DATA.pdf, 377 KB” file the best way in.

    Kind regards,

    Tom

  2. Rex says

    I was looking at the Fn sequence and noticed that even when the digits are flipped they still sum to the Fn sequence .By reversing the digits in a mirror image and subtracting there is a “new “number that is the mirror of a Fn,

    A few of the sums had more calculations than a mirror flip and subtraction.
    For instance 610 flipped is 1006.The digits are intact and “mirrored” like a circle but the sum isn’t flipped.

    The other two involve a negative number which resolves by breaking the number up and subtracting.btw…109 also has a unique Fibonacci connection with 89.

    With the Fn single digits I paired adjacent numbers to form new numbers and they still formed a Fn sequence.

    987,610___789-16 =773 *377*
    610.0,377_1006-773 =*233*
    377,233___773-332 =441 *144*
    233,144___332-441 =-109 *901*_ 90-1=*89*
    144,89 ___441-98 =343 343_ 343-*233*=110_110/2=*55*
    89,55_____98-55 =43 *34*
    55,34_____55-43 =12 *21*
    34,21_____43-12 =31 *13*
    21,13_____12-31 =-19 91 9-1=*8*
    13,8______31-8 =23 *3+2* =5
    8,5,3,2____85-32 =53 *3+5* =8
    5,3,2,1____53-21 =32 *2+3* =5
    3 2,1,1____32-11 =21 *1+2* =3
    2,1,1,0____21-10 =11 *1+1* =2
    1,1,0_____11-10 =1 *1+0 =1
    1,0 ______1-0 =1 1+0 =1

  3. João Café says

    Hello,

    Very interesting indeed. I was trying to apply the Fibonacci series in a musical composition and observed the pattern. The conversion from number to musical note was just subtracting octaves (12) until the number is in [1,13] interval. Do you think this reduction is correct, for musical purposes?

    Regards

    • says

      The concept make sense, but you might want to give it a bit more range for a more pleasing, realistic musical interpretation. It simply reduces every note to sit within a single octave while most songs have a range of low to high that covers more than an octave. An alternate approach would be to extend your range of allowable notes to cover two octaves, with a rule that would not allow any note to be more than an octave than the one before it.

  4. Sergio Viana says

    This pattern is a 12 pattern and it’s opposite…
    The polyhedron with 12 faces is a DODECAHEDRON

    Plato said in his writings that the DODECAHEDRON is the shape of the entire cosmos

    N.A.S.A discovered in 2003 that the shape of the Universe is a DODECAHEDRON

    Man made time as we know it was “ACCIDENTALLY” created being 2 cycles of 12 representing hours and minor cycles of 5×12 = 60 for minutes and seconds…
    Every face of the DODECAHEDRON(12) is a PENTAGON(5) :)

  5. Colin Hetherington says

    My father once ran a number sequence by me that I’ve drifted back to over the past 30 years:

    Start with any two, none zero, digits and note them
    Sum them (if the value is greater sum those two digits ie 13 becomes 4) not result
    Now sum last two digits
    Continue until you get your starting numbers

    This produces 5 Sequences:

    11235843718988764156281911
    13472922461786527977538213
    14595516742685494483257314
    3369663933
    9999

    Obviously, this is simply a numeric doodle based on reduction and the first sequence is simply reduction Fibonacci but I’ve always felt there was something there.

    I’ve stretched this out to 3, 4 and 5 digit reductions (thanks to Excel) and the results show lots of patterns where factors of nine, obviously, run throughout.

    Scaling up the listed article’s proposals for these other sequences is head spinning but strangely beautiful

    I only really reply here as this is the first time I’ve seen my father’s time-passing exercise listed anywhere.

    • wail bourahla says

      I discovered the same thing and also triple figures
      1113598 ………….. 111
      Deceived the 21 series
      Plus five binary strings equals the number of letters of the alphabet
      Please e-mail me back..

  6. JHL says

    Fascinating. Here’s another aspect of the Fibonacci series which reveals another repetitive pattern.

    I just recently discovered that if you take the Fibonacci squares and translate each number to a pitch in the one octave 7-note scale you end up with palindromatic, infinitely repeating series of pitches.

    The first numbers in the Fibonnaci series of squares are 1, 1, 4, 9, 25, 64, 169
    This will translate to the following pitches in one octave: 1, 1, 4, 2 (9-7), 4 (25-21), 1 (64-63), 1 (169-168).

    If we continue this process, this is the pattern we end up with:

    1 1 4 2 4 1 1 7 1 1 4 2 4 1 1 7 etc

    Seeing the 7 as the center you end up with a 15 note palindromatic, infinitely repeating pattern. Not only that, but the pitches between the 7s make up another palindromatic, repeating pattern with the 2 as the center made of 7 pitches (1 1 4 2 4 1 1).

  7. says

    The true 216-digit sequence is (Naturalis Veritas, the end of the history, Massimo Nardotto, 2007):

    1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9
    2 2 4 6 1 7 8 6 5 2 7 9 7 7 5 3 8 2 1 3 4 7 2 9
    3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9
    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
    6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9
    7 7 5 3 8 2 1 3 4 7 2 9 2 2 4 6 1 7 8 6 5 2 7 9
    8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9
    9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

    Ref. http://en.wikipedia.org/wiki/Talk%3APi_(film)

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