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%I #39 Aug 13 2021 09:11:40
%S 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,
%T 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,
%U 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
%N 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) ].
%C Gives number of row in Wythoff array that contains n. - _Casey Mongoven_, Sep 10 2005
%C 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
%H J. H. Conway and N. J. A. Sloane, <a href="/A019586/a019586.pdf">Notes on the Para-Fibonacci and related sequences</a>
%H Casey Mongoven, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from175to192.pdf">Sonification of multiple Fibonacci-related sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%F Says which row of Wythoff array (starting row count at 0) contains n.
%F If delete first occurrence of 0, 1, 2, 3, ... the sequence is unchanged.
%p A019586 := proc(n::posint)
%p local r,c,W ;
%p for r from 1 do
%p for c from 1 do
%p W := A035513(r,c) ;
%p if W = n then
%p return r-1 ;
%p elif W > n then
%p break ;
%p end if;
%p end do:
%p end do:
%p end proc:
%p seq(A019586(n),n=1..100) ; # _R. J. Mathar_, Aug 13 2021
%t 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];
%t Flatten[Array[row, 9] - 1] (* _Jean-François Alcover_, Jul 12 2016, after _Clark Kimberling_ *)
%Y Equals A003603(n) - 1.
%Y Cf. Wythoff array: A035513.
%K nonn,nice,easy,eigen
%O 1,6
%A _N. J. A. Sloane_ and _J. H. Conway_
%E _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.
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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