A248973 -id:A248973 - cp's OEIS frontend

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

Gordon Hamilton, Aug 23 2013

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

			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

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

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).

a228474 = subtract 1 . length . a248939_row  -- Reinhard Zumkeller, Oct 20 2014
(C++) #include 
  int A228474(long n) { int c=0, s; for(std::map seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { seen[n]=true; ++c; } return c; } // M. F. Hasler, Mar 18 2019

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

# 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

{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 *)

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]

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

A248939 Table read by rows: row n lists the wrecker ball sequence starting with n, or contains -1 if 0 is never reached.

Original entry on oeis.org

0, 1, 0, 2, 1, -1, -4, 0, 3, 2, 0, 4, 3, 1, -2, 2, -3, -9, -16, -8, -17, -7, -18, -6, 7, 21, 6, -10, -27, -45, -26, -46, -25, -47, -24, 0, 5, 4, 2, -1, 3, -2, -8, -15, -7, -16, -6, -17, -5, 8, 22, 7, -9, -26, -44, -25, -45, -24, -46, -23, 1, 26, 0, 6, 5, 3

Offset: 0

Reinhard Zumkeller, Oct 17 2014

The wrecker ball sequence starting with n is defined by x[1] = n, and while x[k] is nonzero, x[k+1] = x[k] + s(k)*k, where s(k) = sign(x[k]) if the "candidate" x[k] - sign(x[k])*k had occurred earlier as some x[j], j <= k, and s(k) = -sign(x[k]) else. (This means, from x[k] one moves a distance of k towards the direction of 0 if the result did not occur earlier, or else in the opposite direction.) - M. F. Hasler, Mar 18 2019
A228474(n) + 1 gives the length of row n.
It is currently unproved whether all rows are of finite length. In particular, the length of row 11281 is unknown but larger than 32*10^9. - M. F. Hasler, Mar 18 2019
Hans Havermann, running code from Hugo van der Sanden, has found that row 11281 has length 3285983871527. -  N. J. A. Sloane, Mar 22 2019

			0:  0;
1:  1  0;
2:  2  1  -1  -4  0;
3:  3  2  0;
4:  4  3  1  -2  2  -3  -9  -16  -8  -17  -7  -18  -6  7  21  6  -10  -27  -45
  -26  -46  -25  -47  -24  0;
5:  5  4  2  -1  3  -2  -8  -15  -7  -16  -6  -17  -5  8  22  7  -9  -26  -44
  -25  -45  -24  -46  -23  1  26  0;
6:  6  5  3  0;
7:  7  6  4  1  -3  2  -4  3  -5  -14  -24  -13  -1  12  -2  13 29 . . . . . . . 1730  3445  1729  3446  1728  3447  1727  3448  1726  3449  1725  0;
8:  8  7  5  2  -2  3  -3  4  -4  -13  -23  -12  0;
9:  9  8  6  3  -1  4  -2  5  -3  -12  -22  -11  1  14  0.

Reinhard Zumkeller, Rows n = 0..16 of triangle, flattened
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013
Index entries for sequences related to Recamán's sequence

Cf. A228474 (row lengths - 1), A248961 (row sums), A248973 (partial sums per row), A248952 (min per row), A248953 (max per row), A005132 (Recamán).

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

import Data.IntSet (singleton, member, insert)
a248939 n k = a248939_tabf !! n !! k
a248939_tabf = map a248939_row [0..]
a248939_row n = n : wBall 1 n (singleton n) where
   wBall  0  = []
   wBall k x s = y : wBall (k + 1) y (insert y s) where
                     y = x + (if (x - j) `member` s then j else -j)
                     j = k * signum x

row[0] = 0;
row[n_] := Module[{b}, b[0] = n; b[k_] /; b[k-1] > 0 := b[k] = If[AllTrue[ Range[0, k-1], b[k-1] - k != b[#]&], b[k-1] - k, b[k-1] + k]; b[k_] /; b[k-1] < 0 := b[k] = If[AllTrue[Range[0, k-1], b[k-1] + k != b[#]&], b[k-1] + k, b[k-1] - k]; b[k_] /; b[k-1] == 0 := b[k] = 0; Reap[k = 0; While[b[k] != 0, Sow[b[k++]]]; Sow[0]][[2, 1]]];
row /@ Range[0, 16] // Flatten (* Jean-François Alcover, Sep 27 2019 *)

row(n)={my(M=Map(),L=List(),k=0); while(n, k++; listput(L,n); mapput(M, n, 1); my(t=if(n>0, -k, +k)); n+=if(mapisdefined(M, n+t), -t, t)); listput(L,0); Vec(L)}
  for(n=0, 6, print(row(n))) \\ Andrew Howroyd, Mar 01 2018

def A248939_row(n):
    seen = {n}; orbit = [n]; c = 0
    while n != 0:
        ++c; s = c if n>0 else -c; n += s if n-s in seen else -s
        seen.add(n); orbit.append(n)
    return orbit # M. F. Hasler, Mar 18 2019
(C++) #include 
  A248939_row(long n) { int c=0, s; for(std::map seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { std::cout<M. F. Hasler, Mar 18 2019

T(n,0) = n;
There are three cases for k > 0:
case T(n,k-1) > 0:
  if T(n,k-1) - k != T(n,m), for all m=0..k-1 then T(n,k) = T(n,k-1) - k, otherwise T(n,k) = T(n,k-1) + k,
case T(n,k-1) < 0:
  if T(n,k-1) + k != T(n,m), for all m=0..k-1 then T(n,k) = T(n,k-1) + k, otherwise T(n,k) = T(n,k-1) - k,
case T(n,k-1) = 0:
  T(n,k) = 0; row ends, i.e., k = A228474(n).

Escape clause added by N. J. A. Sloane, Apr 19 2019

A248961 Sums of wrecker ball sequences starting with n.

Original entry on oeis.org

0, 1, -2, 5, -292, -241, 14, -437861, -28, -1, 30, 313, -4472, -4223, -2, 55, 3252, -214246256269, -70, -27, 5260887648, 91, -538, -193, -132, -864538549823, -22, 27, 140, 40053, 53088613819206, 86166834699, 86167898716, 86168962733, 86170026754, 49, 204

Offset: 0

Reinhard Zumkeller, Oct 18 2014

sign

a(n) = A248973(n, A228474(n)) = sum of row n in triangle A248939;

a(A000217(n)) = A000330(n).

			a(1) = 1+0 = 1;
a(2) = 2+1-1-4+0 = -2;
a(3) = 3+2+0 = 5;
a(4) = 4+3+1-2+2-3-9-16-8-17-7-18-6+7+21+6-10-27-45-26-46-25-47-24+0 = -292;
a(5) = 5+4+2-1+3-2-8-15-7-16-6-17-5+8+22+7-9-26-44-25-45-24-46-... = -241;
a(6) = 6+5+3+0 = 14;
a(7) = 7+6+4+1-3+2-4+3-5-14-24-13-1+12-2+13+29+46+28+9-11+10-... = -437861;
a(8) = 8+7+5+2-2+3-3+4-4-13-23-12+0 = -28;
a(9) = 9+8+6+3-1+4-2+5-3-12-22-11+1+14+0 = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Cf. A248939, A248973, A228474, A000217, A000330.

import Data.IntSet (singleton, member, insert)
a248961 n = addup 1 n 0 $ singleton n where
   addup  0 sum  = sum
   addup k x sum s = addup (k + 1) y (sum + x) (insert y s) where
                     y = x + (if (x - j) `member` s then j else -j)
                     j = k * signum x
(C++) #include
long A248961(long n) { long c=0, d, S=n; for(std::set A; n; A.insert(n), S += n += A.count(n - (d = n>0 ? c : -c)) ? d : -d) ++c; return S; } // M. F. Hasler, Mar 19 2019

A248961(n,A=[n],c,S=n)={while( n+=sign(n)*if(setsearch(A,n-sign(n)*c+=1), c, -c), A=setunion(A,[n]); S+=n); S} \\ M. F. Hasler, Mar 19 2019

def A248961(n):
  A = {n}; c = 0; S = 0
  while n != 0:
    ++c; s = c if n>0 else -c; n += s if n-s in A else -s; A.add(n); S += n
  return S # M. F. Hasler, Mar 19 2019

cp's OEIS Frontend

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.

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A248939 Table read by rows: row n lists the wrecker ball sequence starting with n, or contains -1 if 0 is never reached.

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Haskell

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A248961 Sums of wrecker ball sequences starting with n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Python