A324677 - OEIS

%I #8 Mar 11 2019 20:45:23

%S 0,0,0,0,0,0,0,15,0,0,0,27,15,14,0,0,26,12,0,0,4,0,0,0,0,34,0,0,0,34,

%T 2,1,13,1,1,0,0,5,1,3,2,1,0,1,2,0,21,23,22,21,20,19,18,0,0,0,25,26,27,

%U 1,26,1,1,1,33,0,0,4,3,2,1,1,1,39,1,0,0,0,0

%N Starting at n, a(n) is the smallest distance from zero among all positions 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.

%H David Nacin, <a href="/A324677/a324677.png">A324677</a>

%H David Nacin, <a href="/A324677/a324677_1.png">A324677(n)/sqrt(n)</a>

%e For n=11, the points visited are 11, 10, 8, 5, 1, -4, 2, -5, 3, -6, 4, -7, -19, -32, -18, -3, 13, 30, 12, 31, 51, 72, 50, 27, 51, 26, 0.  The only position from which we are forced to revisit a spot is 27, which forces a return to 51. Since this is the only position for which this happens, it is the closest position to zero where this happens and thus a(11)=27.

%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([abs(i) for i in d['stuck']])

%Y Cf. A228474, A324660-A324692.

%K nonn

%O 0,8

%A _David Nacin_, Mar 10 2019