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

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

Author

Reinhard Zumkeller, Oct 17 2014

Keywords

• sign
• tabf

Comments

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

Examples

			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.
		

Links

• 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

Crossrefs

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).

Programs

Haskell

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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).

Extensions

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