These two sequences of positive integers are defined as:

 







R
(
1
)



=
1
 
;
 
S
(
1
)
=
2




R
(
n
)



=
R
(
n
−
1
)
+
S
(
n
−
1
)
,

n
>
1.






{\displaystyle {\begin{aligned}R(1)&=1\ ;\ S(1)=2\\R(n)&=R(n-1)+S(n-1),\quad n>1.\end{aligned}}}




The sequence 



S
(
n
)


{\displaystyle S(n)}

 is further defined as the sequence of positive integers not present in 



R
(
n
)


{\displaystyle R(n)}

.

Sequence 



R


{\displaystyle R}

 starts:

   1, 3, 7, 12, 18, ...

Sequence 



S


{\displaystyle S}

 starts:

   2, 4, 5, 6, 8, ...



Task

 Create two functions named ffr and ffs that when given n return R(n) or S(n) respectively.
(Note that R(1) = 1 and S(1) = 2 to avoid off-by-one errors).
 No maximum value for n should be assumed.
 Calculate and show that the first ten values of R are:
 1, 3, 7, 12, 18, 26, 35, 45, 56, and 69
 Calculate and show that the first 40 values of ffr plus the first 960 values of ffs include all the integers from 1 to 1000 exactly once.


References

 Sloane's A005228 and A030124.
 Wolfram MathWorld
 Wikipedia: Hofstadter Figure-Figure sequences.


#include <iomanip>
#include <iostream>
#include <set>
#include <vector>
 
using namespace std;
 
unsigned hofstadter(unsigned rlistSize, unsigned slistSize)
{
    auto n = rlistSize > slistSize ? rlistSize : slistSize;
    auto rlist = new vector<unsigned> { 1, 3, 7 };
    auto slist = new vector<unsigned> { 2, 4, 5, 6 };
    auto list = rlistSize > 0 ? rlist : slist;
    auto target_size = rlistSize > 0 ? rlistSize : slistSize;
 
    while (list->size() > target_size) list->pop_back();
 
    while (list->size() < target_size)
    {
        auto lastIndex = rlist->size() - 1;
        auto lastr = (*rlist)[lastIndex];
        auto r = lastr + (*slist)[lastIndex];
        rlist->push_back(r);
        for (auto s = lastr + 1; s < r && list->size() < target_size;)
            slist->push_back(s++);
    }
 
    auto v = (*list)[n - 1];
    delete rlist;
    delete slist;
    return v;
}
 
ostream& operator<<(ostream& os, const set<unsigned>& s)
{
    cout << '(' << s.size() << "):";
    auto i = 0;
    for (auto c = s.begin(); c != s.end();)
    {
        if (i++ % 20 == 0) os << endl;
        os << setw(5) << *c++;
    }
    return os;
}
 
int main(int argc, const char* argv[])
{
    const auto v1 = atoi(argv[1]);
    const auto v2 = atoi(argv[2]);
    set<unsigned> r, s;
    for (auto n = 1; n <= v2; n++)
    {
        if (n <= v1)
            r.insert(hofstadter(n, 0));
        s.insert(hofstadter(0, n));
    }
    cout << "R" << r << endl;
    cout << "S" << s << endl;
 
    int m = max(*r.rbegin(), *s.rbegin());
    for (auto n = 1; n <= m; n++)
        if (r.count(n) == s.count(n))
            clog << "integer " << n << " either in both or neither set" << endl;
 
    return 0;
}