Fibonacci family

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The Fibonacci family includes all sequences of the form:

F(n) = \begin{cases} X&\textrm{if\ }n=0\\ Y&\textrm{if\ }n=1\\ A \times F(n-2)+B \times F(n-1)&\textrm{if\ }n\geq2 \end{cases}

i.e., the sequences defined by a second-order linear homogeneous recurrence relation.

Contents

[edit] Special cases

[edit] Fibonacci sequence

A special and the most known case is the Fibonacci sequence, where X = 0, Y = 1, and A = B = 1.

[edit] Square root of 2

Set A to 1, and B to 2, and start with 1,3 and 1,2. This gives two sequences.

Taking quotients of corresponding elements we get:

\frac{1}{1},\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29}...

Solving the recurrence relations shows that this sequence has limit \sqrt{2}.

[edit] See also

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