A134625 Sum-fill array starting with (1,2,3,4,...).

1, 2, 1, 3, 3, 1, 4, 2, 4, 1, 5, 5, 3, 5, 1, 6, 7, 5, 4, 6, 1, 7, 4, 2, 7, 5, 7, 1, 8, 9, 7, 3, 9, 6, 8, 1, 9, 11, 12, 8, 4, 11, 7, 9, 1, 10, 6, 11, 2, 11, 5, 13, 8, 10, 1, 11, 13, 13, 9, 7, 14, 6

Offset: 1

Author

Clark Kimberling, Nov 04 2007

Keywords

• nonn
• tabl

Comments

Every row is a permutation of the positive integers. (Row 2) = A006369. The sequence represents the para-sequence in which the "final ordering" << is given by 1 << ... << 4 << 3 << 2. In every row after row n, for each k<=n, k+1 precedes k and all the numbers between k+1 and k exceed k+1.

Examples

			Starting with x = row 1, Step 1 gives
y = (1,3,2,5,3,7,4,9,5,11,6,13,...).
Delete the second 3,5,7,... leaving row 2:
(1,3,2,5,7,4,9,11,6,13,...).
Northwest corner:
1 2 3 4 5 6 7 8
1 3 2 5 7 4 9 11
1 4 3 5 2 7 12 11
1 5 4 7 3 8 2 9
1 6 5 9 4 11 7 10.
		

References

• C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Links

• Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Crossrefs

Cf. A134626, A134627, A134628.

Formula

Row 1 is the sequence of positive integers. Row n>=2 is produced from row n by the sum-fill operation, defined on an arbitrary infinite or finite sequence x = (x(1), x(2), x(3), ...) by the following two steps: Step 1. Form the sequence x(1), x(1)+x(2), x(2), x(2)+x(3), x(3), x(3)+x(4), ...; i.e., fill the space between x(n) and x(n+1) by their sum. Step 2. Delete duplicates; i.e. letting y be the sequence resulting from Step 1, if y(n+h)=y(n) for some h>=1, then delete y(n+h).