Fibonacci Equations

It has been a long time since I posted any new items. In a caravan with no electricity and no mobile phone reception there is no way of having my own computer, not even with wifi.
In the meantime the Australian Museum in Sydney was kind enough to show a display of mine for 2 weeks as part of the Alive! Exhibition on biodiversity, in September.

Summer is coming and the wild oats are up to 8 ft tall. I am having trouble looking after 4 acres of land by myself working only with hand tools. Pink and red bush roses are abundantly in flower after the unusual rains of winter and spring. Chocolate lilies ( yes they do have this perfume) are flowering in grassy woodland remnants.

Mindful of last year's devastating bushfires I am publishing the equations I have found in connection with the Fibonacci Series. If my caravan were to burn down then all my papers and models would be lost. I am only briefly noting them. There is a lot to document and it might be a slow process. So here goes:-

1. F(n) = F(n-1) + F(n-2)....this is well know; it is definition of the series
F30 = F29 + F28=317811 + 196418 = 514229

2. F(n) = 2F(n-2) + F(n-3).......this is used for seashell spirals
F17 = 2F15 + F14 = 2x377 + 233 = 754 + 233 = 987
F20 = 2F18 + F17 = 2x1597 + 987 = 3194 + 987 = 4181
F30 = 2F28 + F27 = 2x196418 + 121393 = 514229

3. F(1) + F(2) + F(3) +.....+F(n) = F(n + 2) - 1....this is sum total
F1 +....F10 = 0+1+1+2+3+5+8+13+21+34 = 88 = 89-1
F1 + F2 + ....F16 = F18 - 1 = 0+1+1+2+3+....+610 = 1596 = 1597-1

4. F(n) = 7F(n-4) - F(n-8)....this came up while working out how to crochet a whirlpool
F14 = 7x F10 - F6 = (7 x 34) - 5 = 238 - 5 = 233 = F14

5. F(n) = 2F(n-4) + 3F(n-3) .... further "unpacking" of the series.

6. F(n) = 5F(n-4) + 3F(n-5).... ditto of above

7. F(n) = 5F(n-6) + 8F(n-5).... ditto
........................................................................................
8. The following are spookily exact especially at the higher numbers, not always so at lowest iterations:-

F(n) / F(n-1)
F8/F7 = 13/8 = 1.625
F10/F9 = 34/21 = 1.6190476
F12/F11 = 89/55 = 1.6181818
F16/F15 = 610/377 = 1.6180371
F18/F17 = 1597/987 = 1.6180344
F20/F19 = 4181/2584 = 1.618034
F30/F29 = 514229/317811 = 1.6180339
F40/F39 = 63245986/39088169 = 1.6180339

..................................................................................

****This is Phi, a "transcedental number", 1.6180339887498
The golden mean is 1:618
or 2:3:5

****There is a most extraordinary website http://www.humanresonance.org/
 The author Alexander Putney has written a book on Phi.  One can read it online for free.

NB.  5/2013  Incredibly, this website has been stolen!  The author had to buy a new web domain.
 It is http://www.human-resonance.org
...................................................................................

F(n)/F(n-2)
F12/F10 = 89/34 = 2.617647
F20/F18 = 4181/1597 = 2.6180338
F40/F38 = 63245986/24157817 = 2.6180339

F(n)/F(n-3)
F12/F9 = 89/21 = 4.2380952
F20/F17 = 4181/ 987 = 4.2360688
F40/F37 = 63245986/14930352 = 4.2360679

F(n)/F(n-4)
F12/F8 = 89/13 = 6.8461538
F20/F16 = 4181/610 = 6.8540983
F40/F36 = 63245986/9227465 = 6.8541019

F(n)/F(n-5)
F20/F15 = 4181/377 = 11.090185
F36/F31 = 9227465/832040 = 11.090169
F40/F35 = 63245986/5702887 = 11.090169

F(n)/F(n-6)
F16/F10 = 610/34 = 17.941176
F26/F20 = 75025/4181 = 17.944271
F40/F34 = 63245986/3524578 = 17.944271

F(n)/F(n-7)
F10/F3 = 34/1 = 34
F11/F4 = 55/2 = 27.5
F17/F10 = 987/34 = 29.029411
F33/F26 = 2178309/75025 = 29.034441
F40/F33 = 63245986/2178309 = 29.034441

F(n)/F(n-10)
F26/F16 = 75025/610 = 122.9918
F36/F26 = 9227465/75025 = 122.99186
F40/F30 = 63245986/514229 = 122.99186

....................................................

F(n)/F(n-20)

F27/F7 = 121393/8 = 15174.125

F30/F10 = 514229/34 = 15124.382

F55/F35 = 86267571272/5702887

divide top and bottom each by 1000, ie 1.

=86267571/5702.887 = 15126.999

F60/F40 =
956722026041/63245986 = 95672203/6324.5986

= 15126.999

F90/F70 = 1779978866004714/117669030460994

= 17799789/1176.69 = 15126.9999

Amazing, isn't it, and elegant?

It is fascinating playing around with these numbers. There is intricate correlation indeed!

Cheerio from Tiiu Vanamois till later.



9. Half F(n) = 9[ 0.5 x 2F(n-8) + F(n-9)] - half F(n-12)
........ for knitting a half clam shell with wavy lips!

10. Half F(n) = 9[ 0.5x2F(n-7) + F(n-8)] - half F(n-13)
for half clam

11. There is also Half F(n) = 9[F(n-6)] - half F(n-12)

0.5 x 2Fn denotes knit 2 together to make 1.
These equations are not quite right yet but they are on the way.

These lines got placed oddly.
One day I might be able to elaborate more fully
but I think it is beyond the scope of a little blog.