A248952 Smallest term in wrecker ball sequence starting with n.

0, 0, -4, 0, -47, -46, 0, -6362, -23, -22, 0, -32, -471, -470, -29, 0, -218, -4843985, -39, -38, -657367, 0, -101, -57, -56, -7609937, -45, -44, 0, -736, -56168428, -3113136, -3113135, -3113134, -3113133, -51, 0, -190, -1213998, -1213997, -495, -62, -61, -60

Offset

0,3

Comments

Starting at n, a(n) is the minimum value reached according to the following rules. On the k-th step (k = 1, 2, 3, ...) move a distance of k in the direction of zero. If the number landed on has been landed on before, move a distance of k away from zero instead. See A228474 and A248939. - David Nacin, Mar 15 2019

It is currently unproved whether all orbits are finite, and therefore unclear whether all a(n) are well defined. In particular, the orbit of n = 11281 is of unknown length, but is certainly greater than 32*10^9. - M. F. Hasler, Mar 18 2019

Links

M. F. Hasler, Table of n, a(n) for n = 0..5000 (first 1000 terms from Reinhard Zumkeller), Mar 19 2019

Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Formula

a(n) = smallest term in row n of triangle A248939;

a(A000217(n)) = 0; a(A014132(n)) < 0.

Example

a(0) = min{0} = 0;
a(1) = min{1,0} = 0;
a(2) = min{2,1,-1,-4,0} = -4;
a(3) = min{3,2,0} = 0;
a(4) = min{4,3,1,-2,2,-3,-9,-16,-8,-17,-7,-18,-6,7,21,6,-10,-27,...} = -47;
a(5) = min{5,4,2,-1,3,-2,-8,-15,-7,-16,-6,-17,-5,8,22,7,-9,-26,...} = -46;
a(6) = min{6,5,3,0} = 0;
a(7) = min{7,6,4,1,-3,2,-4,3,-5,-14,-24,-13,-1,12,-2,13,29,46,...} = -6362;
a(8) = min{8,7,5,2,-2,3,-3,4,-4,-13,-23,-12,0} = -23;
a(9) = min{9,8,6,3,-1,4,-2,5,-3,-12,-22,-11,1,14,0} = -22.

Prog

(Haskell)
import Data.IntSet (singleton, member, insert, findMin, findMax)
a248952 n = a248952_list !! n
(a248952_list, a248953_list) = unzip $
   map (\x -> minmax 1 x $ singleton x) [0..] where
   minmax _ 0 s = (findMin s, findMax s)
   minmax k x s = minmax (k + 1) y (insert y s) where
                         y = x + (if (x - j) `member` s then j else -j)
                         j = k * signum x
(Python)
#This and sequences A324660-A324692 generated by manipulating this trip function
#spots - positions in order with possible repetition
#flee - positions from which we move away from zero with possible repetition
#stuck - positions from which we move to a spot already visited with possible repetition
def trip(n):
    stucklist = list()
    spotsvisited = [n]
    leavingspots = list()
    turn = 0
    forbidden = {n}
    while n != 0:
        turn += 1
        sign = n // abs(n)
        st = sign * turn
        if n - st not in forbidden:
            n = n - st
        else:
            leavingspots.append(n)
            if n + st in forbidden:
                stucklist.append(n)
            n = n + st
        spotsvisited.append(n)
        forbidden.add(n)
    return {'stuck':stucklist, 'spots':spotsvisited,
                'turns':turn, 'flee':leavingspots}
#Actual sequence
def a(n):
    d = trip(n)
    return min(d['spots'])
# David Nacin, Mar 15 2019
(Python)
def A248952(n):
    return min(A248939_row(n)); # M. F. Hasler, Mar 18 2019
(C++) #include<map>
long A248952(long n) { long c=0, s, m=0; for(std::map<long, bool> seen; n; n += seen[n-(s=n>0?c:-c)] ? s:-s) { if(n<m) m=n; seen[n]=true; ++c; } return m; } // M. F. Hasler, Mar 18 2019

Crossrefs

Cf. A228474 (main entry for wrecker ball sequences).

Cf. A248939, A248953, A000217, A014132.

Sequence in context: A221757 A189424 A009371 * A101502 A118440 A247119

Adjacent sequences: A248949 A248950 A248951 * A248953 A248954 A248955

Keyword

sign

Author

Reinhard Zumkeller, Oct 18 2014

Extensions

Edited by M. F. Hasler, Mar 18 2019

Status

approved