A022414 Kim-sums: "Kimberling sums" K_n + K_3.

2, 7, 10, 4, 15, 18, 20, 23, 9, 28, 31, 12, 36, 39, 41, 44, 17, 49, 52, 54, 57, 22, 62, 65, 25, 70, 73, 75, 78, 30, 83, 86, 33, 91, 94, 96, 99, 38, 104, 107, 109, 112, 43, 117, 120, 46, 125, 128, 130, 133, 51, 138, 141, 143, 146, 56, 151, 154, 59, 159, 162, 164, 167

Offset: 0

Author

Marc LeBrun

Keywords

• nonn
• easy

Comments

Let W(i,j) denote the index of that row of the extended Wythoff array (see A035513) that contains the sequence formed by the sum of rows i and j. Then a(n) = W(2,n). - N. J. A. Sloane, Mar 07 2016

References

• J. H. Conway, Posting to Math Fun Mailing List, Dec 02 1996.
• M. LeBrun, Posting to Math Fun Mailing List Jan 10 1997.

Links

• J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, and 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

Cf. A000201, A035513.

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:
A022414 := proc(n)
    if n = 0 then
        2;
    else
            Kisum(n-1,2) ;
    end if;
end proc:
seq(A022414(i),i=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, 2, Kisum[n - 1, 2]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 15 2023, after R. J. Mathar *)