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

0,6

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.

Links

Table of n, a(n) for n=0..77.

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

Cf. A307326.

Sequence in context: A338526 A182703 A354006 * A263390 A231354 A197119

Adjacent sequences: A307223 A307224 A307225 * A307227 A307228 A307229

Keyword

nonn,tabl

Author

Jan Koornstra, Mar 31 2019

Status

approved