A361819 Irregular triangle read by rows where T(n,k) is the distance which number A361660(n,k) moves in the process described in A361642.

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

Offset: 1

Author

Tamas Sandor Nagy, Mar 25 2023

Keywords

• nonn
• tabf

Comments

Number A361660(n,k) moves to the right and then down and T(n,k) counts the steps in both.

All moves are T(n,k) >= 2 steps since a number moves at least one step right and one step down.

Row n has sum A002378(n-1) which is the total steps to move a column down to a row irrespective of the order of movement.

Each row is a palindrome (the same when reversed), since the moves in A361642 are exactly the reverse moves to send its row back to the starting column.

Examples

			Irregular triangle T(n,k) begins:
  n/k     |   1    2    3    4    5    6    7    8    9
  ------------------------------------------------------
  1       |   (empty row)
  2       |   2;
  3       |   3,   3;
  4       |   4,   2,   2,   4;
  5       |   5,   3,   4,   3,   5;
  6       |   6,   4,   2,   3,   3,   2,   4,   6;
  7       |   7,   5,   3,   5,   2,   5,   3,   5,   7;
 ...
		

Crossrefs

Cf. A361642, A361660, A002541 (row lengths), A002378 (row sums).

Programs

MATLAB

function a = A361819( max_row )
    k = 1;
    for r = 2:max_row
        h = zeros(1,r); h(1) = r;
        while max(h) > 1
           j =  find(h == max(h), 1, 'last' );
           m =  find(h < max(h)-1, 1, 'first' );
           a(k) = (m-j) + (h(j)-h(m)) - 1;
           h(j) = h(j) - 1; h(m) = h(m) + 1;
           k = k+1;
        end
    end
end % Thomas Scheuerle, Mar 27 2023