A307226 Triangle read by rows: drop and bounce.

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, 7, 0, 4, 6, 5, 1, 3, 2, 9, 8, 7, 0, 4, 6, 5, 1, 3, 10, 2, 9, 8, 7, 0, 4, 6, 5, 1, 11, 3, 10, 2, 9, 8

Offset: 0

Author

Jan Koornstra, Mar 31 2019

Keywords

• nonn
• tabl

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.

Examples

			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];
  ...
		

Crossrefs

Cf. A307326.

Programs

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

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)