cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A022423 Kim-sums: "Kimberling sums" K_n + K_12.

Original entry on oeis.org

11, 31, 34, 36, 39, 42, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 70, 73, 76, 78, 81, 83, 86, 89, 91, 94, 97, 99, 102, 104, 107, 110, 112, 115, 118, 120, 123, 125, 128, 131, 133, 136, 138, 141, 144, 146, 149, 152, 154, 157, 159, 162, 165, 167, 170, 172, 175, 178, 180
Offset: 0

Views

Author

Marc LeBrun

Keywords

References

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

Links

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:
A022423 := proc(n)
    if n = 0 then
        11;
    else
        Kisum(n-1,11) ;
    end if;
end proc:
seq(A022423(n),n=0..80) ; # R. J. Mathar, Sep 03 2016
  • Mathematica
    Ki[n_, i_] := Ki[n, i] = Module[{phi = (1 + Sqrt[5])/2}, If[i == 0, n, If[i == 1, Floor[(n+1)*phi], 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, 11, Kisum[n-1, 11]];
a /@ Range[0, 58] (* Jean-François Alcover, Mar 29 2020, after R. J. Mathar *)

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

Original entry on oeis.org

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

Views

Author

Marc LeBrun

Keywords

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

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 *)

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

Original entry on oeis.org

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

Views

Author

Marc LeBrun

Keywords

References

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

Links

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 *)
Showing 1-3 of 3 results.

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