## Caching

Our good friend Mr. Smith has improved his programming skills considerably. Currently, he is learning about recursive functions and he programmed his first recursive piece of code. He wrote a simple seven-liner which calculates the *n ^{th}* element in the

*Fibonacci*sequence:

function TfrmFibonacci.FibonacciRecursive(element: int64): int64; begin if element < 3 then Result := 1 else Result := FibonacciRecursive(element - 1) + FibonacciRecursive(element - 2); end;

I will not argue with him—if you look up the definition of a Fibonacci sequence it really looks like it could be perfectly solved with a recursive function.

A sequence of Fibonacci numbers, *F*, is defined with two simple rules:

First two numbers in the sequence are both 1 (

*F*= 1,_{1}*F*= 1),_{2}Every other number in the sequence is the sum of the preceding two (

*F*=_{n}*F*+_{n-1}*F*)._{n-2}

You will also find a different definition of the Fibonacci sequence in the literature, starting with values 0 and 1, but it only differs...