A324677 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.

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

Offset

0,8

Links

Table of n, a(n) for n=0..78.

David Nacin, A324677

David Nacin, A324677(n)/sqrt(n)

Example

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.

Prog

(Python)
#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}
def minorzero(x):
    if x:
        return min(x)
    return 0
#Actual sequence
def a(n):
    d=trip(n)
    return minorzero([abs(i) for i in d['stuck']])

Crossrefs

Cf. A228474, A324660-A324692.

Sequence in context: A290483 A185294 A287285 * A324675 A106239 A271763

Adjacent sequences: A324674 A324675 A324676 * A324678 A324679 A324680

Keyword

nonn

Author

David Nacin, Mar 10 2019

Status

approved