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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 60 Repeating Pattern

The last digit of the numbers in the Fibonacci Sequence form a pattern that repeats after every 60th number:

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1

This pattern can be seen in the following list of the first 72 Fibonacci numbers:

0	0
1	1
2	1
3	2
4	3
5	5
6	8
7	13
8	21
9	34
10	55
11	89
12	144
13	233
14	377
15	610
16	987
17	1,597
18	2,584
19	4,181
20	6,765
21	10,946
22	17,711
23	28,657
24	46,368
25	75,025
26	121,393
27	196,418
28	317,811
29	514,229
30	832,040
31	1,346,269
32	2,178,309
33	3,524,578
34	5,702,887
35	9,227,465
36	14,930,352
37	24,157,817
38	39,088,169
39	63,245,986
40	102,334,155
41	165,580,141
42	267,914,296
43	433,494,437
44	701,408,733
45	1,134,903,170
46	1,836,311,903
47	2,971,215,073
48	4,807,526,976
49	7,778,742,049
50	12,586,269,025
51	20,365,011,074
52	32,951,280,099
53	53,316,291,173
54	86,267,571,272
55	139,583,862,445
56	225,851,433,717
57	365,435,296,162
58	591,286,729,879
59	956,722,026,041
60	1,548,008,755,920
61	2,504,730,781,961
62	4,052,739,537,881
63	6,557,470,319,842
64	10,610,209,857,723
65	17,167,680,177,565
66	27,777,890,035,288
67	44,945,570,212,853
68	72,723,460,248,141
69	117,669,030,460,994
70	190,392,490,709,135
71	308,061,521,170,129
72	498,454,011,879,264

Lucien Khan arranged these 60 digits of the pattern in a circle, as shown in illustration below:

Here he found other interesting results:

The zeros align with the 4 cardinal points on a compass.
The fives align with the 8 other points of the 12 points on a clock.
Except for the zeros, the number directly opposite each number adds to 10.

Lucien postulates that ancient knowledge of these relationships contributed to the development of our modern use of 60 minutes in an hour, and presentation of numbers on the face of the clock.

I found too that any group of four numbers that are 90 degrees from each other (15 away from each other in the circle) sum to 20, except again for the zeros. As an example, use 1, 7, 9 and 3, which appear one to the right of each of the compass points.

Additionally, every group of five numbers that define the points of the 12 pentagons on the circle also create a pattern. Four of the pentagons have even-numbered last digits of 0, 2, 4, 6, and 8. The remaining eight pentagons have odd-numbered last digits of 1, 3, 5, 7 and 9.

Another interesting pattern yet was observed by Lucien Khan: The 216th number is this sequence is 619220451666590135228675387863297874269396512. The sum of all the digits in that number add up to 216, as well. He notes that it is believed that the secret or hidden name of God contains 216 characters. There are many other fascinating relationships and sacred geometries, which are presented by Lucien Khan in more detail at the links below.

https://www.goldennumber.net/fibonacci-60-repeating-pattern/