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

0,1

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

Table of n, a(n) for n=0..62.

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

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 *)

Crossrefs

Cf. A000201, A035513.

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

Sequence in context: A041453 A042157 A012937 * A319932 A236243 A024831

Adjacent sequences: A022411 A022412 A022413 * A022415 A022416 A022417

Keyword

nonn,easy

Author

Marc LeBrun

Status

approved