OFFSET
0,4
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
LINKS
Reinhard Zumkeller, Rows n = 0..16 of triangle, flattened
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013
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).
EXAMPLE
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.
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 *)
PROG
(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
(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 <bits/stdc++.h>
A248939_row(long n) { int c=0, s; for(std::map<long, bool> seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { std::cout<<n<<", "; seen[n]=true; ++c; } std::cout<<0; } // M. F. Hasler, Mar 18 2019
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Reinhard Zumkeller, Oct 17 2014
EXTENSIONS
Escape clause added by N. J. A. Sloane, Apr 19 2019
STATUS
approved