A019586 - OEIS

%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