%I #6 Feb 02 2022 10:36:43

%S 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,

%T 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

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

%C 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.

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

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Kimberling/kimber16.html"> Self-Containing Sequences, Selection Functions, and Parasequences</a>, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

%F 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).

%e Starting with x = row 1, Step 1 gives

%e y = (1,3,2,5,3,7,4,9,5,11,6,13,...).

%e Delete the second 3,5,7,... leaving row 2:

%e (1,3,2,5,7,4,9,11,6,13,...).

%e Northwest corner:

%e 1 2 3 4 5 6 7 8

%e 1 3 2 5 7 4 9 11

%e 1 4 3 5 2 7 12 11

%e 1 5 4 7 3 8 2 9

%e 1 6 5 9 4 11 7 10.

%Y Cf. A134626, A134627, A134628.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Nov 04 2007