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

1,2

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.

References

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

Links

Table of n, a(n) for n=1..62.

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

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).

Example

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.

Crossrefs

Cf. A134626, A134627, A134628.

Sequence in context: A195915 A219158 A049834 * A325477 A277227 A054531

Adjacent sequences: A134622 A134623 A134624 * A134626 A134627 A134628

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 04 2007

Status

approved