A228474 Number of steps required to reach zero in the wrecker ball sequence starting with n: On the k-th step (k = 1, 2, 3, ...) move a distance of k in the direction of zero. If the result has occurred before, move a distance of k away from zero instead. Set a(n) = -1 if 0 is never reached.

0, 1, 4, 2, 24, 26, 3, 1725, 12, 14, 4, 26, 123, 125, 15, 5, 119, 781802, 20, 22, 132896, 6, 51, 29, 31, 1220793, 23, 25, 7, 429, 8869123, 532009, 532007, 532009, 532011, 26, 8, 94, 213355, 213353, 248, 33, 31, 33, 1000, 9, 144, 110, 112, 82, 84, 210, 60, 34

Offset: 0

Author

Gordon Hamilton, Aug 23 2013

Keywords

• walk
• nonn
• look
• hear

Comments

This is a Recamán-like sequence (cf. A005132).

The n-th triangular number A000217(n) has a(A000217(n)) = n.

a(n) + 1 = length of row n in tables A248939 and A248973. - Reinhard Zumkeller, Oct 20 2014

Hans Havermann, running code from Hugo van der Sanden, has found that a(11281) is 3285983871526. - N. J. A. Sloane, Mar 22 2019

If a(n) != -1 then floor((a(n)-1)/2)+n is odd. - Robert Gerbicz, Mar 28 2019

Examples

			a(2) = 4 because 2 -> 1 -> -1 -> -4 -> 0.
See A248940 for the full 1725-term trajectory of 7. See A248941 for a bigger example, which shows the start of the 701802-term trajectory of 17. - _N. J. A. Sloane_, Mar 07 2019
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 0..10000
• Gordon Hamilton, Wrecker Ball Sequences, Video, 2013. [The sequence is mentioned about 4.5 minutes in to the video. The video begins by discussing A005132. - _N. J. A. Sloane_, Apr 25 2019]
• Hans Havermann, Table of n, a(n) for n = 0..36617
• Hans Havermann, Log-plot of terms through n = 36617
• Hans Havermann, Sharp peaks and high plateaus An overview of large knowns and unknowns up to index 10^6.
• Index entries for sequences related to Recamán's sequence

Crossrefs

Cf. A248939 (rows = the full sequences), A248961 (row sums), A248973 (partial sums per row), A248952 (min per row), A248953 (max per row).

Cf. also A248940 (row 7), A248941 (row 17), A248942 (row 20).

Cf. also A000217, A005132 (Recamán).

Programs

Haskell

a228474 = subtract 1 . length . a248939_row  -- Reinhard Zumkeller, Oct 20 2014
(C++) #include 

Julia

function A228474(n)
    k, position, beenhere = 0, n, [n]
    while position != 0
        k += 1
        step = position > 0 ? k : -k
        position += (position - step) in beenhere ? step : -step
        push!(beenhere, position)
    end
    return length(beenhere) - 1
end
println([A228474(n) for n in 0:16]) # Peter Luschny, Mar 24 2019

Maple

# To compute at most the first M steps of the trajectory of n:
f:=proc(n) local M,i,j,traj,h;
  M:=200; traj:=[n]; h:=n; s:=1;
  for i from 1 to M do j:=h-s*i;
    if member(j,traj,'p') then s:=-s; fi;
    h:=h-s*i; traj:=[op(traj),h];
    if h=0 then return("steps, trajectory =", i, traj); fi;
    s:=sign(h);
  od;
  lprint("trajectory so far = ", traj); error("Need to increase M");
end;  # N. J. A. Sloane, Mar 07 2019

Mathematica

{0}~Join~Array[-1 + Length@ NestWhile[Append[#1, If[FreeQ[#1, #3], #3, Sign[#1[[-1]] ] (Abs[#1[[-1]] ] + #2)]] & @@ {#1, #2, Sign[#1[[-1]] ] (Abs[#1[[-1]] ] - #2)} & @@ {#, Length@ #} &, {#}, Last@ # != 0 &] &, 16] (* Michael De Vlieger, Mar 27 2019 *)

PARI

a(n)={my(M=Map(),k=0); while(n, k++; mapput(M,n,1); my(t=if(n>0, -k, +k)); n+=if(mapisdefined(M,n+t),-t,t)); k} \\ Charles R Greathouse IV, Aug 18 2014, revised Andrew Howroyd, Feb 28 2018 [Warning: requires latest PARI. - N. J. A. Sloane, Mar 09 2019]

Extensions

More terms from Jon E. Schoenfield, Jan 10 2014

Escape clause in definition added by N. J. A. Sloane, Mar 07 2019

Edited by M. F. Hasler, Mar 18 2019