A324682 Starting at n, a(n) is the sign of the closest nonzero point to zero visited for which the next move is a step away from zero, 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. In the case of a tie, or if the set of points is empty, we set a(n)=0.

0, 0, -1, 0, -1, 1, 0, -1, -1, 1, 0, -1, -1, 1, -1, 0, 1, 0, -1, 1, 1, 0, -1, -1, 1, 0, -1, 1, 0, -1, -1, -1, -1, 1, 1, -1, 0, -1, -1, -1, -1, -1, -1, 1, 1, 0, -1, -1, 1, -1, 1, 1, -1, -1, 1, 0, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, 0, -1, 1, 0, -1, -1, -1

Offset

0,1

Links

Table of n, a(n) for n=0..72.

Example

For n=9, the points visited are 9, 8, 6, 3, -1, 4, -2, 5, -3, -12, -22, -11, 1, 14, 0. The three times moves are made away from zero happen at -3, -12 and 1. The closest of these is to zero is 1 and thus a(9) = sgn(1) = 1.

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 sgn(x):
    if x:
        return x//abs(x)
    return 0
def maxorzero(x):
    if x:
        return max(x)
    return 0
def minorzero(x):
    if x:
        return min(x)
    return 0
#Actual sequence
def a(n):
    d = trip(n)
    neg=maxorzero([i for i in d['flee'] if i < 0])
    pos=minorzero([i for i in d['flee'] if i > 0])
    if neg and not pos:
        return -1
    return -sgn(neg+pos)

Crossrefs

Cf. A228474, A324660-A324692.

Sequence in context: A163539 A143538 A242018 * A288132 A324903 A011656

Adjacent sequences: A324679 A324680 A324681 * A324683 A324684 A324685

Keyword

sign

Author

David Nacin, Mar 10 2019

Status

approved