A022416 Kim-sums: "Kimberling sums" K_n + K_5.

4, 13, 16, 18, 21, 24, 26, 29, 31, 34, 37, 39, 42, 45, 47, 50, 52, 55, 58, 60, 63, 65, 68, 71, 73, 76, 79, 81, 84, 86, 89, 92, 94, 97, 100, 102, 105, 107, 110, 113, 115, 118, 120, 123, 126, 128, 131, 134, 136, 139, 141, 144, 147, 149, 152, 154, 157, 160, 162, 165, 168, 170

Offset: 0

Author

Marc LeBrun

Keywords

• nonn

References

• Posting to math-fun mailing list Jan 10 1997.

Programs

Maple

Ki := proc(n,i)
    option remember;
    local phi ;
    phi := (1+sqrt(5))/2 ;
    if i= 0 then
        n;
    elif i=1 then
        floor((n+1)*phi) ;
    else
        procname(n,i-1)+procname(n,i-2) ;
    end if;
end proc:
Kisum := proc(n,m)
    local ks,a,i;
    ks := [seq( Ki(n,i)+Ki(m,i),i=0..5)] ;
    for i from 0 to 2 do
        for a from 0 do
            if Ki(a,0) = ks[i+1] and Ki(a,1) = ks[i+2] then
                return a;
            end if;
            if Ki(a,0) > ks[i+1] then
                break;
            end if;
        end do:
    end do:
end proc:
A022416 := proc(n)
    if n = 0 then
        4;
    else
        Kisum(n-1,4) ;
    end if;
end proc:
seq(A022416(n),n=0..80) ; # R. J. Mathar, Sep 03 2016

Mathematica

Ki[n_, i_] := Ki[n, i] = Which[i == 0, n, i == 1, Floor[(n+1)* GoldenRatio], True, Ki[n, i-1] + Ki[n, i-2]];
Kisum[n_, m_] := Module[{ks,a,i}, ks = Table[Ki[n, i] + Ki[m, i], {i, 0, 5}]; For[i = 0, i <= 2, i++, For[a = 0, True, a++, If[Ki[a, 0] == ks[[i+1]] && Ki[a, 1] == ks[[i+2]], Return[a]]; If[Ki[a, 0] > ks[[i+1]], Break[]]]]];
A022416[n_] := If[n == 0, 4, Kisum[n-1, 4]];
Table[A022416[n], {n, 0, 80}] (* Jean-François Alcover, Jun 09 2023, after R. J. Mathar *)