Generalized Fibonacci Sequence

What is the formula for the generalized Fibonnaci sequence,

$F_{n + k} = F_{n + k - 1} + F_{n + k - 2} + ... + F_{n}$ , where $k \in \mathbb{N}$

#Math

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

You can solve using the characteristic equation (just wiki recurrence relations). Note that the sequence's general formula is dependent on what you choose for F1, F2 ... Fk - based on what the Fi are.

Gabriel Wong - 8 years, 2 months ago

Indeed. Learn how to solve Linear Recurrence Relations using the characteristic equations.

The characteristic equation is $x^k = x^{k-1} + x^{k-2} + \ldots + x^0$ , which has $k$ roots of the form $\{ \alpha_i\}_{i=1}^k$ . Then, your sequence will have the value $F_n = \sum A_i (\alpha_i)^n$ , where $A_i$ depends on the starting values that you chose. [Note: I glossed over the case of repeated roots, which have to be treated differently].

For example, when $k = 2$ , then the equation is $x^2 = x + 1$ or equivalently $x^2 - x - 1=0$ . This has roots $\frac {1 \pm \sqrt{5} } {2}$ , and appear as the powers in Binet's formula.

Calvin Lin Staff - 8 years, 2 months ago

The formula involves the square root of 5, which I write as sqrt[5]. The nth Fibonacci is:

 a*x^n + (1-a)*y^n

where a = (3+sqrt[5])/(5+sqrt[5]) x = (1+ sqrt[5])/2 y = (1- sqrt[5])/2

So my challenges to you are: first, to prove that the formula works, and second, to derive the equation you want for the sum of the first n terms, using the formula for the sum of a geometric series.

superman son - 8 years, 2 months ago

You can use Binet's Formula to find the n-th Fibonacci number. It is easy to prove by induction.

Zi Song Yeoh - 8 years, 2 months ago

I know the formula for the nth Fibonacci number, I'm asking for the formula for the recurrence stated. (including Fibonacci Sequence)

Zi Song Yeoh - 8 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$