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

1,3

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.

Links

Table of n, a(n) for n=1..88.

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

Example

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)

Prog

(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

Crossrefs

Cf. A002260, A220073.

Sequence in context: A267485 A092779 A328067 * A286093 A078827 A157806

Adjacent sequences: A355789 A355790 A355791 * A355793 A355794 A355795

Keyword

nonn,tabl

Author

Tamas Sandor Nagy, Jul 17 2022

Status

approved