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2015-06-06T23:09:08+05:30

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The Fibonacci Sequence is the series ofnumbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the twonumbers before it. Similarly, the 3 is found by adding the two numbers before it.
it always have 0 and 1 in starting of the series.
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2015-06-07T00:22:29+05:30

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Fibonacci numbers:

T_1 = 0,\ \ \ T_2=1\\\\T_n=T_{n-1}+T_{n-2},\ \ \ n \ge 2\\\\So\ \ Fibonacci\ numbers\ are\ \ 0,1,1,2,3,5,8,13,21,34,55,.....

Fn  means the n th term in the sequence.

A term in Fibonacci series is obtained by adding the previous two numbers in the series.  Thus Fibonacci sequence is defined by a recurrence relation.

The sum of the first n fibonacci numbers is equal to the n+2 th term minus 1.
\Sigma_{i=1}^n F_i=F_{n+2} - 1

The series is named after the Italian mathematician  Fibonacci in early 12th century.   It seems that the same sequence was introduced in India earlier to that.  However, that was not made famous sufficiently.    This sequence was associated with Indian mathematicians  Susantha Goonatilake, Virahanka, Gopala, Hemachandra, Pingala (BC).  It appears in Sanskrit Prosody tradition.

Fibonacci numbers are the results of summation of numbers along shallow diagonals in Pascal's triangle (of binomial cofficients). 
F_n\ =\ \Sigma_k\ {}^{(n-k-1)}C_k\ \ \ from\ k =\ 0\ to\ k = \frac{n-1}{2}

Fibinaci numbers are also related to the Golden ratio  (in triangles).
As the number n becomes infinity,    \frac{F_{n+1}}{F_n}  ratio  becomes nearly equal to  the goldren ratio value, 1.618033....

We have Cassin's Identity:          
(F_n)^2 + (-1)^n = F_{n+1} * F_{n-1}

Also,
F_{2n-1} = (F_n)^2 + (F_{n-1})^2\\\\F_{2n} = (F_{n-1} + F_{n+1} ) * F_n

Fibonacci numbers also have  an interesting association with spiral.  If we plot the successive Fibonacci numbers as the radii along x, y, -x, -y , x , y , -x, -y axes in 2-d graph, then we get a spiral.  This is close to the golden spiral. 
 
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