Formulae and Patterns for Fibonacci Numbers in other bases

[gogopoco]

Is there a formula for Fibonacci numbers in other bases, such as hexadecimal or binary? i.e., 55 converted to binary is 00110111. Is there a general formula for that? I'm trying to find patterns in Fibonacci numbers in other bases (binary, octal, or hexadecimal). It is difficult to derive a formula for Fibonacci numbers based on the decimal formula in other bases, primarily because variables cannot be converted. Also, I'm also looking for patterns of Fibonacci numbers in other bases.

[ChrisG]

Hi gogopoco,
That's a creative idea you've got for your project.
The form of the equation for a Fibonacci sequence is the same whether it is in binary, octal, decimal or hexadecimal. The numbers themselves do not change. It is only the symbolic representation of the numbers that changes, depending on the numeral system being used to describe those numbers.
Here is the equation in decimal:
http://en.wikipedia.org/wiki/Fibonacci_number
In binary, it would be the same except that the numerals would be replaced with their binary equivalents. Likewise for octal or hex.
Good luck!
Chris

[gogopoco]

Thanks for your help. Also, I need to prove that F(n) squared + F(n+1) squared = F(2n+1). I know its possible to prove it using Binet's formula, which is Fn = 1/5 ((1+sqrt5)/2) to the power of n - (1-sqrt5)/2) to the power of n). Can you help me prove it?

[ChrisG]

You should be able to use binet's formula to find equations for F(n)^2, F(n+1)^2, and F(2n+1). Then you can substitute these terms into your original equation and then rearrange terms in your expanded equation to show that the left hand side is equal to the right hand side.
Can you please explain more about the overall objectives of this science fair project?
Thanks & good luck!
Chris