Playing with the Number Nine

What if we divide the numbers of the Fibonacci Series by NINE?
4181/9 = 464.55555*rpt
2584/9 = 287.11111*
1597/9 = 177.44444*
..........
34/9 = 3.77777*
21/9 = 2.33333*
13/9 = 1.44444*
A Pattern emerges esp from the last three:-
Fn / 9 = F(n + 5) / 100 plus "extra"...eg .004444* or .003333*
F14 is 377, divide by 9 = 41.88888*
F14 + 5 = F19 = 4181; F19 / 100 = 41.81000- extra is .0788888*
F13 is 233, divide by 9 = 25.88888*
F13 + 5 = F18 = 2584; F18 / 100 = 25.84 plus .04888*
.07888 / .04888 = 1.6137479

Higher number ratios of "extras" = closer to 1.618, the "Golden Mean"
eg F19/9 = 464.555*. F24/100 = 463.68; difference is 0.87555*
F18/9 = 287.111*. F23/100 = 286.57; difference is 0.54111I
ratio of differences is 1.618069
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One can compile tables of the Fibonacci numbers divided by 9; also by 2, 3, 4, 5, 6, 7, 8.
Only one number to F25 divides evenly by all single digits except for 5.
ie number F24 which is 46 368
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What if we multiply the first 25 Fibonacci Numbers by NINE?
I shall only do a few:-
17711 x 9 = 159 399
10946 x 9 = 98 514
6765 x 9 = 60 885
4181 x 9 = 37 629
2584 x 9 = 14 373
987 x 9 = 8 883
610 x 9 = 5 490
377 x 9 = 3 393
233 x 9 = 2 097
144 x 9 = 1 296
89 x 9 = 801
55 x 9 = 495
An Equation emerges:- 9 x Fn = 1/2F(n+6) + 1/2F(n-6) !
eg. 987 x 9 = 1/2 x 17711 + 1/2 x 55
=8855.5 + 27.5 = 8 883 = 987 x 9 !
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Multiply by 18,
then the Equation is 18 x Fn = F(n+6) + F(n-6)

eg. 4181 x 18 = 75258 = 75025 + 233 !
Try it for other numbers, even huge ones.....
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Multiply by 27,
then the Equation is 27 x Fn = 10 x F(n+2) + 1/2F(n+1) + 1/2F(n-8)
eg. 27 x 10946 = 10 x 28657 + 1/2 x 17711 + 1/2 x 233
.........................= 286570 + 8855.5 + 116.5 = 295542
.........................= 27 x 10946 !
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Multiply by 36, (4 x 9).
The Equations might be 36 x Fn = 100F(n-3) + 10 x Fn + F(n+1) + F(n-2) + F(n-4) + F(n-9).....OR......
36 x Fn = 100Fn-3) + 10F(n-3) + F(n+4) + F(n+2) + F(n-1) + F(n-4) + F(n-6) + F(n-10) - F(n-12)

try 233 or 610; I will check on this sometime.....
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What if we Multiply by 45, (5 x 9), ?
The Equations are:-
45 x Fn = F(n+8) - F(n+1) - F(n-3) - F(n-5) - F(n-7)......and.....

45 x Fn = F(n+7) + F(n+5) + F(n+3) + F(n-1) + F(n-8)......then.....

F(n+8) - F(n-8) = F(n+7) + F(n+5) + F(n+3) + F(n+1) + F(n-1) + F(n-3) + F(n-5) + F(n-7) !
try 610.....
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Something might happen with x54, x63, x72, x81, etc; I have not tried it yet...
What use is it? It is just a buzz to see Patterns, " palindromes" almost!
There might be a particular model which one can make with these patterns.....
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What if we multiply by 19?
An Equation that emerges is:-
19 x Fn = F(n+6) + Fn + F(n_6) !
eg. 19 x 610 = 11 590 = 10946 + 610 + 34, correct.
Try other numbers; see how far up the series one can go with this!
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What if we multiply by 29?
The Equation is 29 x Fn = F(n+7) - F(n-7) !
eg. 29 x F18 = F25 - F11
......29 x 2584 = 74936 = 75025 - 89, = correct.
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Multiply by 39:-
Equation seems to be 39 x Fn = 1/2 F(n+9) + Fn - 1/2 F(n-9) !
eg. 39 x 610 = 46368/2 + 610 - 8/4
.....................=23184 + 610 - 4 = 23790. = correct.
eg. 39 x 144 = 10946/2 +144 - 2/2
.....................=5473 + 144 - 1 = 5616, = correct.
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Multiply by 59 or 99:-
there could well be Patterns; I am working on it.....
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At least I have got it all down on paper and online
There could well be corrections to any mistakes...
At least it satisfies my curiosity.
What does the number nine signify indeed?!

For the Maya, the number nine was to be worshipped.
Nine "Lords of the Night" were painted on the tomb walls of the Temple of Inscriptions.
1/9 = 0.1111*rpt
2/9 = 0.2222
3/9 = 0.3333
....
8/9 = 0.8888
....
10/9 = 1.1111
20/9 = 2.2222rpt
nine based decimal system is half as complex as today's 10 based system,
This is from page 324 of " The Mayan Prophecies"
by Adrian G. Gilbert and Maurice M Cotterell publ 1995 Element Books, Inc.


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You might like to play with the first 135 Fibonacci numbers:-
See post 5/4/2013

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