

A267700


"Tree" sequence in a 90degree sector of the cellular automaton of A160720.


22



0, 1, 2, 5, 6, 9, 12, 19, 20, 23, 26, 33, 36, 43, 50, 65, 66, 69, 72, 79, 82, 89, 96, 111, 114, 121, 128, 143, 150, 165, 180, 211, 212, 215, 218, 225, 228, 235, 242, 257, 260, 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
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OFFSET

0,3


COMMENTS

Conjecture: this is also the "tree" sequence in a 120degree sector of the cellular automaton of A266532.
It appears that this is also the partial sums of A038573.
a(n) is also the total number of ON cells after nth stage in the tree that arises from one of the four spokes in a 90degree sector of the cellular automaton A160720 on the square grid.
Note that the structure of A160720 is also the "outward" version of the UlamWarburton cellular automaton of A147562.
It appears that A038573 gives the number of cells turned ON at nth stage.
Conjecture: a(n) is also the total number of Ytoothpicks after nth stage in the tree that arises from one of the three spokes in a 120degree sector of the cellular automaton of A266532 on the triangular grid.
Note that the structure of A266532 is also the "outward" version of the Ytoothpick cellular automaton of A160120.
It appears that A038573 also gives the number of Ytoothpicks added at nth stage.
Comment from N. J. A. Sloane, Jan 23 2016: All the above conjectures are true!
a(n) is also the number of nondecreasing binarycontainment 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:
(1,1) (1,1) (1,1) (1,1) (1,1) (1,1)
(2,2) (1,3) (1,3) (1,3) (1,3)
(2,2) (2,2) (1,5) (1,5)
(2,3) (2,3) (2,2) (2,2)
(3,3) (3,3) (2,3) (2,3)
(4,4) (3,3) (2,6)
(4,4) (3,3)
(4,5) (4,4)
(5,5) (4,5)
(4,6)
(5,5)
(6,6)
(End)


LINKS



FORMULA

Conjecture 1: a(n) = (A266532(n+1)  1)/3.


MATHEMATICA

Accumulate[Table[2^DigitCount[n, 2, 1]1, {n, 0, 30}]] (* based on conjecture confirmed by Sloane, Gus Wiseman, Mar 31 2019 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



