A022415 Kim-sums: "Kimberling sums" K_n + K_4.

3, 10, 13, 15, 18, 21, 23, 26, 28, 31, 34, 36, 39, 42, 44, 47, 49, 52, 55, 57, 60, 62, 65, 68, 70, 73, 76, 78, 81, 83, 86, 89, 91, 94, 97, 99, 102, 104, 107, 110, 112, 115, 117, 120, 123, 125, 128, 131, 133, 136, 138, 141, 144, 146, 149, 151, 154, 157, 159, 162, 165, 167, 170, 172, 175, 178

Offset: 0

Author

Marc LeBrun

Keywords

• nonn

References

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

Links

• J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, On Kimberling Sums and Para-Fibonacci Sequences, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997

Crossrefs

The "Kim-sums" K_n + K_i for i = 2 through 12 are given in A022413, A022414, A022415, A022416, ..., A022423.

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:
A022415 := proc(n)
    if n = 0 then
        3;
    else
        Kisum(n-1,3) ;
    end if;
end proc:
seq(A022415(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[]]]]];
a[n_] := If[n == 0, 3, Kisum[n - 1, 3]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 15 2023, after R. J. Mathar *)