A355792 Triangular array, read by rows. The rules of the construction are described in the Comments section.

1, 1, 2, 2, 3, 1, 1, 2, 4, 3, 3, 1, 2, 4, 5, 5, 6, 3, 1, 2, 4, 4, 5, 6, 3, 1, 7, 2, 2, 4, 5, 8, 6, 3, 1, 7, 7, 2, 4, 5, 8, 6, 3, 1, 9, 9, 10, 7, 2, 4, 5, 8, 6, 3, 1, 1, 9, 11, 10, 7, 2, 4, 5, 8, 6, 3, 3, 1, 9, 11, 12, 10, 7, 2, 4, 5, 8, 6, 6, 3, 1, 9, 11, 12, 10, 13, 7, 2

Offset: 1

Author

Tamas Sandor Nagy, Jul 17 2022

Keywords

• nonn
• tabl

Comments

Row m contains m distinct elements of the set {1..m}. The last element k(m) in row m becomes the first one in row m+1. There, the remaining elements of row m repeat in the same order, with element m+1 inserted immediately after that which is the k(m)-th in row m.

Examples

			Triangle begins
  1;
  1,  2;
  2,  3,  1;
  1,  2,  4,  3;
  3,  1,  2,  4,  5;
  5,  6,  3,  1,  2,  4;
  4,  5,  6,  3,  1,  7,  2;
  2,  4,  5,  8,  6,  3,  1,  7;
  7,  2,  4,  5,  8,  6,  3,  1,  9;
  9, 10,  7,  2,  4,  5,  8,  6,  3,  1;
  1,  9, 11, 10,  7,  2,  4,  5,  8,  6,  3;
  3,  1,  9, 11, 12, 10,  7,  2,  4,  5,  8,  6;
  6,  3,  1,  9, 11, 12, 10, 13,  7,  2,  4,  5,  8;
  ...
To illustrate the rule:
Row 6 ends with 4, therefore the next row, row 7, begins with 4.
The order of the rest of the elements in row 6, that is, 5, 6, 3, 1, and 2, remains unchanged in row 7, while there the new element 7 is introduced immediately after 1 since the 4th element in row 6 is 1.
From _Jon E. Schoenfield_, Jul 17 2022: (Start)
The diagram below illustrates the way in which, on each row, each number from the previous row is placed either to the left or the right of the new number (which is identified by parentheses):
.
                                             (1)
.
                                            1  (2)
                                             \
                                      2  (3)  1
                                     /     \
                                1   2  (4)  3
                               /   /   /
                          3   1   2   4  (5)
                           \   \   \   \
                    5  (6)  3   1   2   4
                   /   /   /   /     \
              4   5   6   3   1  (7)  2
             /   /     \   \   \   \
        2   4   5  (8)  6   3   1   7
       /   /   /   /   /   /   /
  7   2   4   5   8   6   3   1  (9)
(End)
		

Crossrefs

Cf. A002260, A220073.

Programs

MATLAB

function a = A355792( max_row )
    T = cell(1,1); T{1} = 1;
    for n = 2:max_row
        j = mod(T{n-1}(end),n-1);
        s = circshift(T{n-1},1,2);
        T{n} = [s(1:j+1) n s(j+2:end)];
    end
    a = [T{1:end}];
end % Thomas Scheuerle, Jul 18 2022

Formula

From Jon E. Schoenfield, Jul 17 2022: (Start)

T(1, 1) = 1.

For n > 1, let j = 2 + (T(n-1, n-1) mod (n-1)); then

T(n, k) = T(n-1, n-1) if k = 1

= T(n-1, k-1) if 1 < k < j

= n if k = j

= T(n-1, k-2) otherwise. (End)

Let b(n) = max(a(1),a(2),...,a(n)) then -(1/2) < 2^(1/2)*n^(1/2)-b(n) < (1/2). - Thomas Scheuerle, Jul 18 2022