A324686 Starting at n, a(n) is the number of times we move away from zero from a negative position 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, 1, 0, 8, 8, 0, 409, 2, 2, 0, 2, 31, 31, 2, 0, 15, 201444, 3, 3, 25388, 0, 10, 5, 5, 347717, 3, 3, 0, 75, 874341, 103653, 103652, 103652, 103652, 3, 0, 10, 52376, 52375, 48, 5, 4, 4, 251, 0, 32, 23, 23, 16, 16, 44, 10, 4, 4, 0, 113, 112, 111, 129093

Offset

0,5

Links

Table of n, a(n) for n=0..59.

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.  Of these, we move away from zero at -7, -19, 13, 12, 31, 51, and 27. Two of these are negative and thus a(11)=2.

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)
    return sum(1 for i in d['flee'] if i<0)

Crossrefs

Cf. A228474, A324660-A324692.

Sequence in context: A019960 A069527 A154492 * A155840 A373657 A365707

Adjacent sequences: A324683 A324684 A324685 * A324687 A324688 A324689

Keyword

nonn

Author

David Nacin, Mar 10 2019

Status

approved