A134626 Sum-fill array starting with (1,2,4,8,16,...), powers of 2.

1, 2, 1, 4, 3, 1, 8, 2, 4, 1, 16, 6, 3, 5, 1, 32, 4, 5, 4, 6, 1, 64, 12, 2, 7, 5, 7, 1, 128, 8, 8, 3, 9, 6, 8, 1, 256, 24, 6, 8, 4, 11, 7, 9, 1, 512, 16, 10, 2, 11, 5, 13, 8, 10, 1, 1024, 48, 16, 10, 7, 14, 6

Offset

1,2

Comments

(Row 2) is possibly A074323 except for an initial 1. The sequence represents the para-sequence in which the "final ordering" << is given by 1 << ... << 4 << 3 << 2. Row n contains 1,2,3,...2n, but not 2n+1. Row n starts like row n of A134625; e.g., row 6 of A123625 and row of A134626 have the same first 16 terms.

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 A000079. 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 3, Step 1 gives
y = (1,5,4,7,3,8,5,7,2,10,8,14,6,...).
Delete the second 5,7,8,... leaving row 4:
(1,5,4,7,3,8,2,10,14,6,...).
Northwest corner:
1 2 4 8 16 32
1 3 2 6 4 12
1 4 3 5 2 8
1 5 4 7 3 8
1 6 5 9 4 11.

Crossrefs

Cf. A134625, A134627, A134628.

Sequence in context: A210213 A305695 A211235 * A347633 A115450 A109435

Adjacent sequences: A134623 A134624 A134625 * A134627 A134628 A134629

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 04 2007

Status

approved