Dear Antony,
I was playing with Fibonacci series this evening while sitting in my backyard and came across two other series of numbers. I will put down how I arrived at them.
I wrote the Fibonacci series on a paper up to 12th degree on either side of 0 as follows
-144 89 -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55 89 144
I virtually folded the paper in my mind at 0 so that the numbers on either side overlapped and added them to each other where they aligned.
I got a new series
0 2 0 4 0 10 0 26 0 68 0 178 0 466 0 1220 0 3194
ignoring the 0's it read as 0 2 4 10 26 68 178 466 ....
Soon I realized that this can be defined by an equation
Sn = 2 * Sn-1 sigma (I=2 to n) Sn-i
with the seeds of
S0 = 0
S1 = 2
S2 = 2 * S1 S0 = 2 * 2 0 = 4
S3 = 2 * S2 S1 S0 = 2* 4 2 0 = 10
S4 = 2 * S3 S2 S1 S0 = 2 * 10 4 2 0 = 26
I also found that division of the two successive numbers soon converges on 2.618 which happens to be the square of golden ratio 1.618.
Now I went back to the original Fibonacci series and virtually folded it at 0 in my mind again and this time I subtracted the numbers where they aligned and I got another series as
0 2 0 6 0 16 0 42 0 110 0 288
Ignoring the 0's this read as 0 2 6 16 42 110 288 and I realized that this can be defined as an equation as well
Sn = 3 * Sn-1 - Sn-2
with S0 = 0 and S1 = 2 as the seeds
S2 = 3 * S1 - S0 = 3 * 2 - 0 = 6
S3 = 3 * S2 - S1 = 3 * 6 - 2 = 16
S4 = 3 * S3 - S2 = 3 * 16 - 6 = 42
S5 = 3 * S4 - S3 = 3 * 42 - 16 = 110
I also found that division of the two successive numbers in this series also soon converges on 2.618 which happens to be the square of golden ratio 1.618.
Finally I did another interesting thing, merged these two series and got another one which read as
First series --->0 2 0 4 0 10 0 26 0 68 0 178 0 466
Second Series ---> 0 2 0 6 0 16 0 42 0 110 0 288 0
Merged series -->0 2 2 4 6 10 16 26 42 68 110 178 288 466
I realized that the merged series is a new Fibonacci type series with a different second seed of 2 instead of 1. Even this series successive number division yields the golden ratio of 1.618.
Now I asked my self if we can have 2 as a second seed and produce another series which yields the same golden ratio why not 3 and soon found that
0 3 3 6 9 15 24 39 63 102 .... is also a series that also converges on the golden ratio 1.618.
0 4 4 8 12 20 32 52 84 136...... is also a series that also converges on the golden ratio 1.618
So any Fibonacci type series with 0 as the first seed and I ( from 1 to infinity) as the second seed will have the successive numbers ration in them converging on the golden ratio of 1.618. This again proves the point that I, the singularity, is equally the same everywhere. Mathematics is pointing to the absoluteness of I in Fibonacci too.
Love,
Sridattadev.
0 2 2 4 6 10 16 26