# Lucas number

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The Lucas series is the sequence of numbers:

```2, 1, 3, 4, 7, 11, 18, 29, 47, …
```

The idea is, that the next number is produced by summing the two preceding ones.

## generation

### recursive implementation

``` 3 type
4 	/// domain for Lucas number function
5 	/// where result fits within a nativeUInt
6 	// You can not name it lucasDomain,
7 	// since the Lucas number function itself
8 	// is defined for all whole numbers
9 	// but the result beyond L(n) exceeds high(nativeUInt).
10 	lucasLeftInverseRange =
11 		{\$ifdef CPU64} 0..92 {\$else} 0..46 {\$endif};
12
13 {**
14 	calculates Lucas number recursively
15
16 	\param n the index of the Lucas number to calculate
17 	\return the Lucas number at n,
18 	        unless n is out of range, then 0
19 }
20 function lucas(const n: lucasLeftInverseRange): nativeUInt;
21 begin
22 	case n of
23 		// recursive case
24 		2..high(n):
25 		begin
26 			lucas := lucas(n - 2) + lucas(n - 1);
27 		end;
28 		// base cases
29 		1:
30 		begin
31 			lucas := 1;
32 		end;
33 		0:
34 		begin
35 			lucas := 2;
36 		end;
37 		// abort case
38 		otherwise
39 		begin
40 			// neutral element of addition
41 			// indicating n is out of range
42 			// [there is no n satisfying L(n) = 0]
43 			lucas := 0;
44 		end;
45 	end;
46 end;
```

### based on Fibonacci sequence

The Lucas numbers can be calculated by using the `fibonacci` function shown in the article Fibonacci number.

```54 {**
55 	calculates Lucas number based on Fibonacci numbers
56
57 	\param n the index of the Lucas number to calculate
58 	\return the Lucas number at n,
59 	        unless n is out of range, then 0
60 }
61 function lucas(const n: lucasLeftInverseRange): nativeUInt;
62 begin
63 	case n of
64 		1..high(n):
65 		begin
66 			lucas := fibonacci(n - 1) + fibonacci(n + 1);
67 		end;
68 		0:
69 		begin
70 			// We can not deduce L(0) from Fibonacci
71 			// since that function is not defined for negative n
72 			// [we would call fibonacci(-1) + fibonacci(1)].
73 			lucas := 2;
74 		end;
75 		otherwise
76 		begin
77 			// neutral element of addition
78 			// indicating n is out of range
79 			// [there is no n satisfying L(n) = 0]
80 			lucas := 0;
81 		end;
82 	end;
83 end;
```

### iterative implementation

This is in line with the iterative implementation of `fibonacci` but with differing start values. Therefore the code is not repeated here.

## lookup

Calculating Lucas numbers every time they are needed is time-consuming. While the code required for generation takes almost no space, a lookup table is the means of choice when an application needs them a lot. Since Lucas numbers can be derived from Fibonacci numbers, usually only one table (those with the Fibonacci series) is stored, being a compromise between efficiency and memory utilization. In order to use the `fibonacci` table without margin case treatment, it has to be at least expanded to `fibonacci[-1]`.

An actual implementation is omitted here, since everyone wants it differently. Also, do not program, what's already been programmed. For instance, the Lucas number functions of the GNU multiple precision arithmetic library take this approach.