Fibonacci Sequence Digital Root Pattern

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Patterns in the Digital Roots of the Fibonacci Sequence

Another attribute of the Fibonacci Sequence is the repeating pattern sequence of 24 numbers [1] of the Digital Roots[2] of the Sequence. The Digital Root is found by adding together all of the digits of a number to obtain a single digit.

But wait! There's more!

If the sequence of the 24 repeating digital root values is divided into two groups of twelve, then each pair of roots in sequence between the two groups add to nine. Note that the last pair or roots add to 18, which has a digital root of nine.

Note: Repeating patterns in modulo sequences is not unique to the Fibonacci or Lucas sequences alone. Nearly any sequence generated by a regular process will produce patterns at some modulo.

 n  Fibonacci
Sequence
Digital 
Root
 n   Fibonacci
Sequence
Digital
Root
Root of
Root Sums
1 1 1 13 233 8 9
2 1 1 14 377 8 9
3 2 2 15 610 7 9
4 3 3 16 987 6 9
5 5 5 17 1597 4 9
6 8 8 18 2584 1 9
7 13 4 19 4181 5 9
8 21 3 20 6765 6 9
9 34 7 21 10946 2 9
10 55 1 22 17711 8 9
11 89 8 23 28657 1 9
12 144 9 24 46368 9 9
First 24 Fibonacci number digital roots in pairs 12 apart mapped on a nonagon of nine sides. Fibonacci digital roots repeat every 24 numbers of the sequence, and the 1st 12 and 2nd 12 add consecutively to nine. Created with Inkscape 0.47. © 2010 DonEMitchell

Gibonacci Sequence

Card Colm by Colm Mulcahy
Gibonacci Bracelets
Card tricks based on moduli of a Generalized Fibonacci Sequence (Gibonacci Sequence)

Pack of 13 Spheres

illustrations pending

There are 12 vertices of a cuboctahedron, which is an octahedron truncated at the midpoints of its edges.

The vertices of a cuboctahedron define four planes of anti-symmetry.

Six vertices occupy a plane, and there are four of these planes. 6 X 4 = 24.

Each vertex occupies two planes.

See also

Patterns in the Fibonacci Numbers by Dr. R. Knott

References

  1. Lost Secrets of the Phi Code, Jain (http://news.branyvnimani.cz/index.php?article_id=10131)
  2. Digital Root, MathWorld.Wolfram.com(http://mathworld.wolfram.com/DigitalRoot.html)