OFFSET
0,3
COMMENTS
Conjecture: this is also the "tree" sequence in a 120-degree 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 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.
Note that the structure of A160720 is also the "outward" version of the Ulam-Warburton cellular automaton of A147562.
It appears that A038573 gives the number of cells turned ON at n-th stage.
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.
Note that the structure of A266532 is also the "outward" version of the Y-toothpick cellular automaton of A160120.
It appears that A038573 also gives the number of Y-toothpicks added at n-th stage.
Comment from N. J. A. Sloane, Jan 23 2016: All the above conjectures are true!
From Gus Wiseman, Mar 31 2019: (Start)
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:
(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
a(n) = (A160720(n+1) - 1)/4.
Conjecture 1: a(n) = (A266532(n+1) - 1)/3.
All of the above conjectures are true. - N. J. A. Sloane, Jan 23 2016
(Conjecture) a(n) = A267610(n) + n. - Gus Wiseman, Mar 31 2019
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
Omar E. Pol, Jan 19 2016
STATUS
approved