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

%I A134625 #6 Feb 02 2022 10:36:43
%S A134625 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 A134625 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 A134625 Sum-fill array starting with (1,2,3,4,...).
%C A134625 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 A134625 C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.
%H A134625 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 A134625 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 A134625 Starting with x = row 1, Step 1 gives
%e A134625 y = (1,3,2,5,3,7,4,9,5,11,6,13,...).
%e A134625 Delete the second 3,5,7,... leaving row 2:
%e A134625 (1,3,2,5,7,4,9,11,6,13,...).
%e A134625 Northwest corner:
%e A134625 1 2 3 4 5 6 7 8
%e A134625 1 3 2 5 7 4 9 11
%e A134625 1 4 3 5 2 7 12 11
%e A134625 1 5 4 7 3 8 2 9
%e A134625 1 6 5 9 4 11 7 10.
%Y A134625 Cf. A134626, A134627, A134628.
%K A134625 nonn,tabl
%O A134625 1,2
%A A134625 _Clark Kimberling_, Nov 04 2007