Formula for Fibonacci sequence with arbitrary initial numbers

• Final formula for Fibonacci sequences with arbitrary initial numbers is established.
• Length of Fibonacci spiral for initial numbers 0 and 1 is calculated.
• Relation between hyperbolic and trigonometric cosine is established.
• Complex form of Fibonacci numbers is suggested.

In this paper the formula for Fibonacci sequences with arbitrary initial numbers has been established by using damped oscillation equation. The formula has an exponential and an oscillatory part, it does not separate the indexes of odd and even members of the series and it is applicable on the continual domain. With elementary conditions the formula is reduced to Lucas series, and the square of Lucas series has a catalytic role in the relation of hyperbolic and trigonometric cosine. A complex function is given and the length of Fibonacci spiral is calculated. Natural phenomena support the validity of the proposed concept.