OFFSET
0,6
COMMENTS
Number k starts with 1, 2, 3, ... 'points' and is 'dropped' on the initial value of 0. k then 'bounces' on the numbers already in the sequence according to the following rules:
- k loses 1 point for each bounce.
- After the first bounce (i.e., on 0) k moves to the right. If k bounces on a number already in the sequence which has a higher value than k, then the direction reverses. At lower or equal values, k continues to move in the same direction.
- If k reaches the start or the end of the sequence or has 0 points left, the starting value of k is added to the sequence at that position.
EXAMPLE
For k = 4:
4 points, bounce on 0, move to the right;
3 points, bounce on 1, move to the right;
2 points, bounce on 3, reverse direction;
1 point, bounce on 1, move to the left;
0 points: k = 4 is inserted between 0 and 1. (Although equal to the next value 0, k has no more points left to bounce over it.)
The first iterations:
[0];
[0, 1];
[0, 1, 2];
[0, 1, 3, 2];
[0, 4, 1, 3, 2];
[0, 4, 5, 1, 3, 2];
[0, 4, 6, 5, 1, 3, 2];
[7, 0, 4, 6, 5, 1, 3, 2];
[7, 0, 4, 6, 5, 1, 3, 2, 8];
...
PROG
(Python)
seq = [0]
for k in range (1, 10):
points = k
position = seq.index(0)
direction = 1
while points > 0 and position >= 0 and position < len(seq):
if points < seq[position]: direction *= -1
points -= 1
position += direction
else:
if position < 0: seq.insert(0, k)
elif position == len(seq): seq.append(k)
elif points == 0 and direction == 1: seq.insert(position, k)
else: seq.insert(position - direction, k)
print(seq)
(PARI) process(row, n) = {my(pos = 1, dir = 1, m = n); while (m && (pos >= 1) && (pos <= #row), if (m < row[pos], dir = -dir); pos += dir; m--; ); if (pos == 0, return (concat(n, row))); if (pos == #row +1, return (concat(row, n))); if (dir == -1, pos ++); my(nrow = vector(#row+1)); for (k=1, pos-1, nrow[k] = row[k]; ); nrow[pos] = n; for (k = pos+1, #row+1, nrow[k] = row[k-1]; ); return (nrow); }
tabl(nn) = {row = [0]; print(row); for (n=1, nn, row = process(row, n); print(row); ); } \\ Michel Marcus, Apr 13 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jan Koornstra, Mar 31 2019
STATUS
approved