# Fibonacci number

The Fibonacci Sequence is the series of numbers:

``` 0, 1, 1, 2, 3, 5, 8, 13, 21, …
```

The idea is to add the two last numbers in order to produce the next value.

## generation

The following implementations show the principle of how to calculate Fibonacci numbers. They lack of input checks. Depending on your preferences you might either want to generate a run-time error (e. g. by `{\$rangeChecks on}` or utilizing `system.runError`), raise an exception, or simply return a bogus value indicating something went wrong.

### recursive implementation

``` 3type
4	/// domain for Fibonacci function
5	/// where result is within nativeUInt
6	// You can not name it fibonacciDomain,
7	// since the Fibonacci function itself
8	// is defined for all whole numbers
9	// but the result beyond F(n) exceeds high(nativeUInt).
10	fibonacciLeftInverseRange =
11		{\$ifdef CPU64} 0..93 {\$else} 0..47 {\$endif};
12
13{**
14	implements Fibonacci sequence recursively
15
16	\param n the index of the Fibonacci number to retrieve
17	\returns the Fibonacci value at n
18}
19function fibonacci(const n: fibonacciLeftInverseRange): nativeUInt;
20begin
21	// optimization: then part gets executed most of the time
22	if n > 1 then
23	begin
24		fibonacci := fibonacci(n - 2) + fibonacci(n - 1);
25	end
26	else
27	begin
28		// since the domain is restricted to non-negative integers
29		// we can bluntly assign the result to n
30		fibonacci := n;
31	end;
32end;
```

### iterative implementation

This one is preferable for its run-time behavior.

```13{**
14	implements Fibonacci sequence iteratively
15
16	\param n the index of the Fibonacci number to calculate
17	\returns the Fibonacci value at n
18}
19function fibonacci(const n: fibonacciLeftInverseRange): nativeUInt;
20type
21	/// more meaningful identifiers than simple integers
22	relativePosition = (previous, current, next);
23var
24	/// temporary iterator variable
25	i: longword;
26	/// holds preceding fibonacci values
27	f: array[relativePosition] of nativeUInt;
28begin
29	f[previous] := 0;
30	f[current] := 1;
31
32	// note, in Pascal for-loop-limits are inclusive
33	for i := 1 to n do
34	begin
35		f[next] := f[previous] + f[current];
36		f[previous] := f[current];
37		f[current] := f[next];
38	end;
39
40	// assign to previous, bc f[current] = f[next] for next iteration
41	fibonacci := f[previous];
42end;
```

## lookup

While calculating the Fibonacci number every time it is needed requires almost no space, it takes a split second of time. Applications heavily relying on Fibonacci numbers definitely want to use a lookup table instead. And yet in general, do not calculate what is already a known fact. Since the Fibonacci sequence doesn’t change, actually calculating it is a textbook demonstration but not intended for production use.

Nevertheless an actual implementation is omitted here, since everyone wants to have it differently, a different flavor.