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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 numbers	1	1	2	3	5	8	4	3	7	1	8	9
2nd 12 numbers	8	8	7	6	4	1	5	6	2	8	1	9
Numeric reduction – Add rows 1 and 2	9	9	9	9	9	9	9	9	9	9	9	18
Final numeric reduction – Add digits of result	9	9	9	9	9	9	9	9	9	9	9	9

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
Fibonacci Number	1st Level	2nd Level	Final Level
Example: 2,584	2+5+8+4=19	1+9=10	1+0=1
0	0	0	0
1	1	1	1
1	1	1	1
2	2	2	2
3	3	3	3
5	5	5	5
8	8	8	8
13	4	4	4
21	3	3	3
34	7	7	7
55	10	1	1
89	17	8	8
144	9	9	9
233	8	8	8
377	17	8	8
610	7	7	7
987	24	6	6
1,597	22	4	4
2,584	19	10	1
4,181	14	5	5
6,765	24	6	6
10,946	20	2	2
17,711	17	8	8
28,657	28	10	1
46,368	27	9	9
75,025	19	10	1
121,393	19	10	1
196,418	29	11	2
317,811	21	3	3
514,229	23	5	5
832,040	17	8	8
1,346,269	31	4	4
2,178,309	30	3	3
3,524,578	34	7	7
5,702,887	37	10	1
9,227,465	35	8	8
14,930,352	27	9	9
24,157,817	35	8	8
39,088,169	44	8	8
63,245,986	43	7	7
102,334,155	24	6	6
165,580,141	31	4	4
267,914,296	46	10	1
433,494,437	41	5	5
701,408,733	33	6	6
1,134,903,170	29	11	2
1,836,311,903	35	8	8
2,971,215,073	37	10	1
4,807,526,976	54	9	9
7,778,742,049	55	10	1
12,586,269,025	46	10	1
20,365,011,074	29	11	2
32,951,280,099	48	12	3
53,316,291,173	41	5	5
86,267,571,272	53	8	8
139,583,862,445	58	13	4
225,851,433,717	48	12	3
365,435,296,162	52	7	7
591,286,729,879	73	10	1
956,722,026,041	44	8	8
1,548,008,755,920	54	9	9
2,504,730,781,961	53	8	8
4,052,739,537,881	62	8	8
6,557,470,319,842	61	7	7
10,610,209,857,723	51	6	6
17,167,680,177,565	67	13	4
27,777,890,035,288	73	10	1
44,945,570,212,853	59	14	5
72,723,460,248,141	51	6	6
117,669,030,460,994	65	11	2
190,392,490,709,135	62	8	8
308,061,521,170,129	46	10	1
498,454,011,879,264	72	9	9

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.

Filed Under: Math

https://www.goldennumber.net/fibonacci-24-pattern/

Fibonacci Fun

Here are some illustrations by myself and others, dealing with the Fibonacci sequence and the Golden Number, Phi.

A few months ago, I noticed the exponents of Phi encode the Fibonacci sequence.

The sequence gets closer to the golden number the larger the numbers get. This will work with any two starting numbers, like say 91 and 432, for example. You’ll find the same ratio emerge if you keep adding them together. (91+432 = 523, then, 432+523=955, etc) Already we are at 1.8 to 1. (955/523) A handful more steps and we’ll have Phi to more than 99.999% accuracy.

???? If you had a million phalanges, your hand would look like King Midas.

Jain108 made a post on how, if you start with 2 and 10, you arrive at the actual numbers found in the exponents of Phi. Remember exponents are simply the number multiplied by itself n number of times. Phi cubed, or Phi to the third power, is just Phi x Phi x Phi. (1.618 x 1.618 x 1.618 = 4.236) See that was easy, now stop complaining how you’re bad at math.

Phi to the negative power is the same as saying, one divided by the Phi cubed, or whatever.

Here’s some different ways to say the same thing.

This does an excellent job of explaining where all these freaking numbers come from. (img Scott Onstott)

Circles of Fibonacci. Magenta and green cancel out and make gold

Sam Kutler found the golden number in concentric circles.

The outer ring shows the 24 digit pattern(digital roots), the inner the 60 digit pattern(last number).

Lucien Khan rediscovered the 60 digit repeat pattern.

Fibonacci Patterns

Fibonacci sequence hidden in ordinary division problem:
If you divide 1 by 999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999 (that’s 999 quattuordecillion), the Fibonacci sequence pops out.

Chad Adam’s Extended Fibonacci Series

Like I said, you can start a Fibonacci sequence with any two numbers. If you begin with the numbers 1-9, some interesting things happen.

by Chad Adams

See how the numbers match up?

The Platonic Solids

The Platonic Solids

back to the 60 digit pattern….

The Fibonacci Sequence, starting at the 60th number in the sequence. See the pattern? Just look at the last digit in each number of the sequence. [after 8 is 3 because the last digit in 13(5+8) is 3]