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

Author

Reinhard Zumkeller, Oct 18 2014

Keywords

• sign

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

Examples

			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.
		

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

Crossrefs

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

Cf. A248939, A248953, A000217, A014132.

Programs

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

Formula

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

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

Extensions

Edited by M. F. Hasler, Mar 18 2019