Fibonacci family

The Fibonacci family includes all sequences of the form:

$F(n) = egin{cases}X& extrm{if }n=0\Y& extrm{if }n=1\A imes F(n-2)+B imes F(n-1)& extrm{if }ngeq2end{cases}$

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

Special cases

Fibonacci sequence

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

Square root of 2

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

*1, 3, 7, 17, 41, 99, 239... OEIS|id=A001333
*1, 2, 5, 12, 29, 70, 169... Pell numbers, OEIS|id=A000129

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 $sqrt{2}$ .

See also

* Lucas sequence

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Fibonacci family

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