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

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

%S 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,

%T 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,

%U 16,10,7,14,6

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

%C (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.

%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 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).

%e Starting with x = row 3, Step 1 gives

%e y = (1,5,4,7,3,8,5,7,2,10,8,14,6,...).

%e Delete the second 5,7,8,... leaving row 4:

%e (1,5,4,7,3,8,2,10,14,6,...).

%e Northwest corner:

%e 1 2 4 8 16 32

%e 1 3 2 6 4 12

%e 1 4 3 5 2 8

%e 1 5 4 7 3 8

%e 1 6 5 9 4 11.

%Y Cf. A134625, A134627, A134628.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Nov 04 2007