1,2

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.

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

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).

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.

Cf. A134626, A134627, A134628.

Adjacent sequences: A134622 A134623 A134624 this_sequence A134626 A134627 A134628

Sequence in context: A098975 A127121 A049834 this_sequence A054531 A115131 A117895

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Nov 04 2007