A324678 - OEIS

%I #5 Mar 11 2019 20:45:30

%S 0,0,0,0,0,0,0,-3165,0,0,0,0,-140,-139,0,0,0,-3072845,0,0,-383171,0,0,

%T 0,0,-4869724,0,0,0,-217,-31071367,-1854085,-1854084,-1854083,

%U -1854082,0,0,-24,-696919,-696918,-26,-1,0,0,-1920,0,-148,-86,-85,-84,-83,-144

%N Starting at n, a(n) is the minimum negative position from which a spot must be revisited on the next move, or zero if no such positions exist, 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.

%e For n=41, the points visited are 41, 40, 38, 35, 31, 26, 20, 13, 5, -4, 6, -5, 7, -6, 8, -7, 9, -8, 10, -9, 11, -10, 12, -11, -35, -60, -34, -61, -33, -62, -32, -1, -33, 0.  The only position from which we are forced to revisit a spot is -1 which forces a return to -33.  As this is the only position and it is negative, it is the minimum negative position and thus a(41)=-1.

%o (Python)

%o #Sequences A324660-A324692 generated by manipulating this trip function

%o #spots - positions in order with possible repetition

%o #flee - positions from which we move away from zero with possible repetition

%o #stuck - positions from which we move to a spot already visited with possible repetition

%o def trip(n):

%o     stucklist = list()

%o     spotsvisited = [n]

%o     leavingspots = list()

%o     turn = 0

%o     forbidden = {n}

%o     while n != 0:

%o         turn += 1

%o         sign = n // abs(n)

%o         st = sign * turn

%o         if n - st not in forbidden:

%o             n = n - st

%o         else:

%o             leavingspots.append(n)

%o             if n + st in forbidden:

%o                 stucklist.append(n)

%o             n = n + st

%o         spotsvisited.append(n)

%o         forbidden.add(n)

%o     return {'stuck':stucklist, 'spots':spotsvisited,

%o                 'turns':turn, 'flee':leavingspots}

%o def minorzero(x):

%o     if x:

%o         return min(x)

%o     return 0

%o #Actual sequence

%o def a(n):

%o     d=trip(n)

%o     return minorzero([i for i in d['stuck'] if i<0])

%Y Cf. A228474, A324660-A324692.

%K sign

%O 0,8

%A _David Nacin_, Mar 10 2019