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%I #58 Oct 10 2022 09:37:46
%S 0,1,2,5,6,9,12,19,20,23,26,33,36,43,50,65,66,69,72,79,82,89,96,111,
%T 114,121,128,143,150,165,180,211,212,215,218,225,228,235,242,257,260,
%U 267,274,289,296,311,326,357,360,367,374,389,396,411,426,457,464,479,494,525,540,571,602,665,666,669,672,679,682,689
%N "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.
%C Conjecture: this is also the "tree" sequence in a 120-degree sector of the cellular automaton of A266532.
%C It appears that this is also the partial sums of A038573.
%C a(n) is also the total number of ON cells after n-th stage in the tree that arises from one of the four spokes in a 90-degree sector of the cellular automaton A160720 on the square grid.
%C Note that the structure of A160720 is also the "outward" version of the Ulam-Warburton cellular automaton of A147562.
%C It appears that A038573 gives the number of cells turned ON at n-th stage.
%C Conjecture: a(n) is also the total number of Y-toothpicks after n-th stage in the tree that arises from one of the three spokes in a 120-degree sector of the cellular automaton of A266532 on the triangular grid.
%C Note that the structure of A266532 is also the "outward" version of the Y-toothpick cellular automaton of A160120.
%C It appears that A038573 also gives the number of Y-toothpicks added at n-th stage.
%C Comment from _N. J. A. Sloane_, Jan 23 2016: All the above conjectures are true!
%C From _Gus Wiseman_, Mar 31 2019: (Start)
%C a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:
%C (1,1) (1,1) (1,1) (1,1) (1,1) (1,1)
%C (2,2) (1,3) (1,3) (1,3) (1,3)
%C (2,2) (2,2) (1,5) (1,5)
%C (2,3) (2,3) (2,2) (2,2)
%C (3,3) (3,3) (2,3) (2,3)
%C (4,4) (3,3) (2,6)
%C (4,4) (3,3)
%C (4,5) (4,4)
%C (5,5) (4,5)
%C (4,6)
%C (5,5)
%C (6,6)
%C (End)
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F a(n) = (A160720(n+1) - 1)/4.
%F Conjecture 1: a(n) = (A266532(n+1) - 1)/3.
%F Conjecture 2: a(n) = A160720(n+1) - A266532(n+1).
%F All of the above conjectures are true. - _N. J. A. Sloane_, Jan 23 2016
%F (Conjecture) a(n) = A267610(n) + n. - _Gus Wiseman_, Mar 31 2019
%t Accumulate[Table[2^DigitCount[n,2,1]-1,{n,0,30}]] (* based on conjecture confirmed by Sloane, _Gus Wiseman_, Mar 31 2019 *)
%Y Cf. A000120, A038573, A147562, A160120, A160720, A161336, A169779, A266532, A266534, A266536.
%Y Cf. A006218, A019565, A070939, A080572, A267610, A267700.
%Y Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325124.
%K nonn
%O 0,3
%A _Omar E. Pol_, Jan 19 2016
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