FIBONACCI Numbers Formulae

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Other Pages : PELL Numbers Formulae ,  Sums of Powers,  FIBONACCI Numbers Formulae,  TRIANGLE Numbers that are Perfect Squares, Sums of Cubes, Sums of Powers - Ramanujan and his Number 1729

A brief introduction to Fibonacci Numbers : 1,1,2,3,5,8,13,21,34,55,89,....... which is given by

F(k+2) = F(k+1) + F(k),  F(0)=0, F(1) =1

For more information on Fibonacci Numbers please see Dr. Ron Knott's excellent site at  http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

The following is a list of Fibonacci Formulae that I have found. 

The following are the notations used :
1) SIGMA(m=1 to k) = Sum from m=1 to m=k
2) NCR(n,r) = n Choose r = n! / (r! (n-r)!)
3) [n] = integer part of n, for example [2.5] = 2
4) a^b = a to the power of b
5) a*b = a multiplied by b. Note that I have used * only to separate terms and not for every case like 2k = 2*k
6) |a| = absolute value of a, for example |-1| = 1

Here is a very interesting formula involving e, i and PI :
 

F(k) = SIGMA(m=1 to k) 5^((k-1)/2)*e^( i*((m-1)/2)*Pi  + i*(k-m)*arctan(-2))* NCR(2k-m, m-1)

which is the same as

F(k) = SIGMA(m=1 to k) i^(m-1)*(1 - 2i)^(k-m)*NCR(2k-m, m-1)

 FeiPI

 

F(k) = SIGMA(m=1 to [(k+1)/2]) NCR(k-m, m-1) F1
F(2k) = SIGMA(m=1 to [(k+1)/2]) (-1)^(m-1)*3^(k-2m+1)*NCR(k-m, m-1) F2
F(2k) = SIGMA(m=1 to k) (-1)^(m-1)*5^(k-m)*NCR(2k-m, m-1) F3
F(4k) = 3 * SIGMA(m=1 to [(k+1)/2]) (-1)^(m-1)*7^(k-2m+1)*NCR(k-m, m-1) F4
F(4k) = 3 * SIGMA(m=1 to k) 5^(k-m)*NCR(2k-m, m-1) F5
F(k)^2 = SIGMA(m=1 to k) SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k-m-j, m-j) F6
F(2k)^2 = SIGMA(m=1 to k) (-1)^(m-1) * 3^(2k-2m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k-m-j, m-j) F7
F(2k)^2 = SIGMA(m=1 to 2k) (-1)^(m-1) * 5^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j, m-j) F8
F(4k)^2 = 9 * SIGMA(m=1 to k) (-1)^(m-1) * 7^(2k-2m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k-m-j, m-j) F9
F(4k)^2 = 9 * SIGMA(m=1 to 2k) 5^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j, m-j) F10
SIGMA F(k)^2 = SIGMA(m=1 to k) SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k+1-m-j, m-j) F11
|SIGMA (-1)^(k-1) * F(2k)^2| = SIGMA(m=1 to k) (-1)^(m-1) * 9^(k-m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k+1-m-j, m-j) F12
SIGMA F(2k) = SIGMA(m=1 to k) 3^(k-m) * SIGMA(j=1 to [(m+1)/2] ) (-1)^(j-1)*NCR(k-m+j-1, j-1) F13
SIGMA F(2k) = SIGMA(m=1 to k) (-1)^(m-1) * 5^(k-m) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(2k+1-m-j, m-j) F14
(F(2k) + F(2k-2))^2 = SIGMA(m=1 to 2k-1) (-1)^(m-1) * 5^(2k-m-1) * SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j-2, m-j) F15
(F(2k) - F(2k-2))^2 = F(2k-1))^2 = SIGMA(m=1 to 2k-1) SIGMA(j=1 to m) (-1)^(j-1)*NCR(4k-m-j-2, m-j) F16