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Triangle read by rows: drop and bounce.
1

%I #53 Jun 07 2019 11:13:12

%S 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,

%T 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,

%U 4,6,5,1,11,3,10,2,9,8

%N Triangle read by rows: drop and bounce.

%C 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:

%C - k loses 1 point for each bounce.

%C - 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.

%C - 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.

%e For k = 4:

%e 4 points, bounce on 0, move to the right;

%e 3 points, bounce on 1, move to the right;

%e 2 points, bounce on 3, reverse direction;

%e 1 point, bounce on 1, move to the left;

%e 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.)

%e The first iterations:

%e [0];

%e [0, 1];

%e [0, 1, 2];

%e [0, 1, 3, 2];

%e [0, 4, 1, 3, 2];

%e [0, 4, 5, 1, 3, 2];

%e [0, 4, 6, 5, 1, 3, 2];

%e [7, 0, 4, 6, 5, 1, 3, 2];

%e [7, 0, 4, 6, 5, 1, 3, 2, 8];

%e ...

%o (Python)

%o seq = [0]

%o for k in range (1, 10):

%o points = k

%o position = seq.index(0)

%o direction = 1

%o while points > 0 and position >= 0 and position < len(seq):

%o if points < seq[position]: direction *= -1

%o points -= 1

%o position += direction

%o else:

%o if position < 0: seq.insert(0, k)

%o elif position == len(seq): seq.append(k)

%o elif points == 0 and direction == 1: seq.insert(position, k)

%o else: seq.insert(position - direction, k)

%o print(seq)

%o (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);}

%o tabl(nn) = {row = [0]; print(row); for (n=1, nn, row = process(row, n); print(row););} \\ _Michel Marcus_, Apr 13 2019

%Y Cf. A307326.

%K nonn,tabl

%O 0,6

%A _Jan Koornstra_, Mar 31 2019