Lucas number
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English (en) │
suomi (fi) │
français (fr) │
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 previous ones.
generation
The following implementations intends to just show the principle. They lack of input checking, thus has uncertain behavior when supplied with parameters out of range.
recursive implementation
3type
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}
19function lucas(const n: lucasLeftInverseRange): nativeUInt;
20begin
21 case n of
22 2..high(n):
23 begin
24 lucas := lucas(n - 2) + lucas(n - 1);
25 end;
26 1:
27 begin
28 lucas := 1;
29 end;
30 0:
31 begin
32 lucas := 2;
33 end;
34 otherwise
35 begin
36 lucas := 0;
37 end;
38 end;
39end;
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 numbersü
56
57 \param n the index of the Lucas number to calculate
58 \return the Lucas number at n
59}
60function lucas(const n: lucasLeftInverseRange): nativeUInt;
61begin
62 if n > 0 then
63 begin
64 lucas := fibonacci(n - 1) + fibonacci(n + 1);
65 end
66 else
67 begin
68 // L(0) := 2
69 lucas := 2;
70 end;
71end;
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.
An actual implementation is omitted here, since everyone wants it differently.
