FORMULAS FOR FIBONACCI NUMBERS
Fibonacci numbers have long been a subject of study for mathematicians, and even a source of entertainment. Back in school, I was also fascinated by this curious object. As with all my similar research, I wrote down the formulas I derived for Fibonacci numbers in a special notebook.
This page contains excerpts from that notebook. Perhaps others besides me will find this interesting too...
Let's start with the fact that the Fibonacci series is a sequence of numbers, each of which is calculated based on the previous ones. In general:
Ns(n) = Ns(n-1) + Ns(n-1-s)
Where s is the "coverage length", and n is the serial number of the number in the series (N), and s≥0.
It is assumed that Ns(1)=1, and all N for n<1 are equal to 0.
When s=0 the series is an actual power of two:
N = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,...
For s=1 we have a regular Fibonacci series, in which each subsequent number is the sum of the two previous ones:
N = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...
When s=2, this is the sum of the previous and the one 2 positions away:
N = 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41,...
When s=3, this is the sum of the previous and the one 3 positions away:
N = 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, ...
Numbers and the series themselves have interesting properties. Strictly speaking, "Fibonacci numbers" often refer not to the numbers themselves in the series (here, it's simply called the "Fibonacci series"), but to the numbers that are the limit of the ratio of adjacent numbers in the series:
Ns(n+1)
Fs = lim ---------
n→∞ Ns(n)
This limit is finite and completely definable. The values of F are calculated from the equation: "Fss+1 - Fss - 1 = 0"
For s=0 we get "F01 - F00 - 1 = 0", from which F0=2 (that is obvious).
For s=1 we get a quadratic equation: "F12 - F11 - 1 = 0". The positive root of this equation is the most famous Fibonacci number (the "Golden Ratio"):
_
1+\/5
F1 = ----- = 1.618034
2
For s=2, we obtain a cubic equation: "F23 - F22 - 1 = 0". The real root of this equation is the number "1.465571", which is calculated purely analytically as follows:
__________ __________
3 / __ 3 / __
\/116+12*\/93 + \/116-12*\/93 + 2
F2 = ---------------------------------
6
As is known, analytical methods of solution exist for equations of degree no higher than 4. To calculate F3 we have an equation of degree 4: "F34 - F33 - 1 = 0".
The desired root F3=1.380277569. Its analytical expression no longer fits into one formula:
_______________ _______________
3 / _______ 3 / _______
y = \ / \/283/108 + 1/2 - \ / \/283/108 - 1/2
\/ \/
____________________
____ / ____ ____
F3 = (1 + \/1-4y + \/2+4y+2\/1-4y +8\/y2+4 )/4
Fibonacci numbers have quite a wide range of applications, but there's plenty of material about them online. I've only posted here what I once derived myself.
In that era, there was no internet or home computers. All calculations and derivations were done by hand on paper, and the calculations were performed on engineering calculators.