Lucas number

2, 1, 3, 4, 7, 11, 18, 29, 47, ???

type
    /// domain for Lucas number function
    /// where result fits within a nativeUInt
    // You can not name it lucasDomain,
    // since the Lucas number function itself
    // is defined for all whole numbers
    // but the result beyond L(n) exceeds high(nativeUInt).
    lucasLeftInverseRange =
        {$ifdef CPU64} 0..92 {$else} 0..46 {$endif};

{**
    calculates Lucas number recursively
 
    \param n the index of the Lucas number to calculate
    \return the Lucas number at n,
            unless n is out of range, then 0
}
function lucas(const n: lucasLeftInverseRange): nativeUInt;
begin
    case n of
        // recursive case
        2..high(n):
        begin
            lucas := lucas(n - 2) + lucas(n - 1);
        end;
        // base cases
        1:
        begin
            lucas := 1;
        end;
        0:
        begin
            lucas := 2;
        end;
        // abort case
        otherwise
        begin
            // neutral element of addition
            // indicating n is out of range
            // [there is no n satisfying L(n) = 0]
            lucas := 0;
        end;
    end;
end;

{**
    calculates Lucas number based on Fibonacci numbers
 
    \param n the index of the Lucas number to calculate
    \return the Lucas number at n,
            unless n is out of range, then 0
}
function lucas(const n: lucasLeftInverseRange): nativeUInt;
begin
    case n of
        1..high(n):
        begin
            lucas := fibonacci(n - 1) + fibonacci(n + 1);
        end;
        0:
        begin
            // We can not deduce L(0) from Fibonacci
            // since that function is not defined for negative n
            // [we would call fibonacci(-1) + fibonacci(1)].
            lucas := 2;
        end;
        otherwise
        begin
            // neutral element of addition
            // indicating n is out of range
            // [there is no n satisfying L(n) = 0]
            lucas := 0;
        end;
    end;
end;