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Essentially the first differences of A161336.

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)

The structure is the same as A161330 but without the two central E-toothpicks. All terms are multiples of 6.

It appears that if n >> 1 the structure looks like an equilateral triangle, which is essentially one of the six wedges of the E-toothpick (or snowflake) structure of A161330. The sequence gives the number of E-toothpicks in the structure after n stages. A220498 (the first differences) gives the number added at the n-th round. For more information and some illustrations see A161330. For the E-toothpick right triangle see A211964.

The structure looks like a tree which arises from one of the four spokes of the structure of the cellular automaton of A151895.

a(n) is the total number of ON cells after n-th stage.

For n >> 1 the structure looks like a square which is rotated 45 degrees.

First differs from A161336 (snowflake tree) at a(16).

First differs from A266536 at a(13). - Omar E. Pol, Apr 02 2016

The structure looks like a tree which arises from one of the four spokes of the structure of the cellular automaton of A170896.

a(n) is the total number of ON cells after n-th stage.

For n >> 1 the structure looks like a square which is rotated 45 degrees.

First differs from both A161336 (snowflake tree) and A266534 at a(13).

The structure looks like a tree which arises from one of the four spokes of the structure of the cellular automaton of A267190.

a(n) is the total number of ON cells after n-th stage.

For n >> 1 the structure looks like a square which is rotated 45 degrees.

First differs from A161336 at a(17), where A161336 is a version of A161330 (the snowflake cellular automaton).

First differs from A266534 at a(16), where A266534 is a version of A151895.

First differs from A266536 at a(13), where A266536 is a version of A170896 (the Schrandt-Ulam cellular automaton).

From Omar E. Pol, Oct 16 2017: (Start)

The graph of both A266536 and this sequence are very similar.

For n >> 1, it appears that A266534(n) < A161336(n) < a(n) < A266536(n).

The graphs of these four sequences are similar, and the behavior looks like percolation.

It appears that there are no recurrences in these four sequences. Thus it appears that there are no recurrences in A151895, A161330, A267190 and A170896. (End)