A019586 Vertical para-Fibonacci sequence: takes value i on later (i.e., b_j, j >= 2) terms of i-th Fibonacci sequence defined by b_0 = i, b_1 = [ tau(i+1) ].

0, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 6, 1, 7, 4, 0, 8, 5, 3, 9, 2, 10, 6, 1, 11, 7, 4, 12, 0, 13, 8, 5, 14, 3, 15, 9, 2, 16, 10, 6, 17, 1, 18, 11, 7, 19, 4, 20, 12, 0, 21, 13, 8, 22, 5, 23, 14, 3, 24, 15, 9, 25, 2, 26, 16, 10, 27, 6, 28, 17, 1, 29, 18, 11, 30, 7, 31, 19, 4, 32, 20, 12

Offset: 1

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• nice
• easy
• eigen

Comments

Gives number of row in Wythoff array that contains n. - Casey Mongoven, Sep 10 2005

For a method of generating this sequence that does not refer to the Wythoff array or Fibonacci numbers, see A003603. - Clark Kimberling, Oct 29 2009

Links

• J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences
• Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• N. J. A. Sloane, Classic Sequences

Crossrefs

Equals A003603(n) - 1.

Cf. Wythoff array: A035513.

Programs

Maple

A019586 := proc(n::posint)
    local r,c,W ;
    for r from 1 do
        for c from 1 do
            W := A035513(r,c) ;
            if W = n then
                return r-1 ;
            elif W > n then
                break ;
            end if;
        end do:
    end do:
end proc:
seq(A019586(n),n=1..100) ; # R. J. Mathar, Aug 13 2021

Mathematica

row[1] = row[2] = {1}; row[n_] := row[n] = Module[{ro, pos, lp, ins}, ro = row[n - 1]; pos = Position[ro, Alternatives @@ Intersection[ro, row[n - 2]]] // Flatten; lp = Length[pos]; ins = Range[lp] + Max[ro]; Do[ro = Insert[ro, ins[[i]], pos[[i]] + i], {i, 1, lp}]; ro];
Flatten[Array[row, 9] - 1] (* Jean-François Alcover, Jul 12 2016, after Clark Kimberling *)

Formula

Says which row of Wythoff array (starting row count at 0) contains n.

If delete first occurrence of 0, 1, 2, 3, ... the sequence is unchanged.

Extensions

Casey Mongoven reports that where the sequence reads 15,9,2,16,10,6,29,1,30,11,7,19,..., the 29 and 30 should be 17 and 18.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003