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 From: BARILOCHENSE6999  (Original message) Sent: 28/10/2019 18:24

# 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 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.

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

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# 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/

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