A324672 Starting at n, a(n) is the sign of the farthest point from zero visited 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 case of a tie, a(n)=0.

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

Offset

0

Comments

The only known value for which there is a tie (up to n=1000) happens at n=89.

Links

Table of n, a(n) for n=0..73.

Example

For n=2, the points visited are 2,1,-1,-4,0. The farthest point from zero is -4, and sgn(-4) = -1, thus a(2) = -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
#Actual sequence
def a(n):
    d = trip(n)
    mx=max(d['spots'])
    mn=min(d['spots'])
    return sgn(mx+mn)

Crossrefs

Cf. A228474, A324660-A324692.

Sequence in context: A070747 A070750 A011598 * A330035 A011599 A011600

Adjacent sequences: A324669 A324670 A324671 * A324673 A324674 A324675

Keyword

sign

Author

David Nacin, Mar 10 2019

Status

approved