The Fibonacci sequence which begins is defined by

(the starting values vary by author).

This is a special case of a more general sequence given by

There are at least three ways to determine such a sequence. I explore those here.

A Fibonacci-style sequence is determined by any of:

- The first four terms:
- The recurrence constants:
- The closed-form constants: where

These are related as follows:

(for )

The derivation of the formulae for is as follows:

so

so

so

so

so

but

so

but

so, identifying coefficients, (and noting symmetry for later)

so

so

so, multiplying by

so, by symmetry

and the formulae for also follow.

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This entry was posted on Sunday, 29 September 2013 at 23:33 and is filed under Contribution, Mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
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