cp's OEIS Frontend

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)

This sequence is related to the Sierpinski triangle and to Gould's sequence A001316. - Omar E. Pol, Jul 23 2009

When written as a irregular triangle in which row lengths are A011782 it appears that right border gives A173033. - Omar E. Pol, Mar 20 2013

On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.

At round 1, we turn ON four cells, forming a square.

The rule for n > 1: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.

Therefore:

At Round 2, we turn ON twelve cells around the square.

At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.

And so on.

For the first differences see the entry A161411.

Shows a fractal behavior similar to the toothpick sequence A139250.

A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4.

When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).

a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009

From Omar E. Pol, Mar 28 2011: (Start)

Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.

Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.)

The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage.

(End)

The structure has a fractal behavior similar to the toothpick sequence A139250.

First differences: A161415, where there is an explicit formula for the n-th term.

For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.

This cellular automata is formed by the concatenation of three Sierpinski triangles, starting from a central vertex. Adjacent polygons are fused. The ON cells are triangles, but we only count after fusion. The sequence gives the number of polygons at the n-th round.

If instead we start from four Sierpinski triangles we get A160720.

A256251 gives the number of cells turned ON at n-th stage.

Note that the number of cells turned ON at n-th stage in each one of its four quadrants is also A006257 (Josephus problem). For more information see A256249.

It appears that this is also a bisection of A256249.

First differs from A169707 at a(13), but both sequences share infinitely many terms. This one is simpler. Compare A169707.

More precisely, a(n)=sum(iA159913(i)), since we want the sequence to start with a(0)=0 and not with A159913(0)=1.

a(n) is also the total number of ON cells after n generations in the outward corner version of the Ulam-Warburton cellular automaton of A147562, and a(n) is also the total number of Y-toothpicks after n generations in the outward corner version of the Y-toothpick structure of A160120. - David Applegate and Omar E. Pol, Jan 24 2016

For the connection with A160720 (the "outward" version of the Ulam-Warburton cellular automaton A147562) see formula section and A267700.

A266533 (the first differences) gives the number of Y-toothpicks added to the structure at n-th stage.

First differs from A160120 at a(9).

First differs from A160715 at a(13).

First differs from A160720 at a(13).

Analog of A160720, but here we are working on the triangular lattice.

The first differences (A256257) gives the number of triangular cells turned ON at every generation.

Also 6 times the sum of all entries in rows 0 to n of Sierpiński's triangle A047999.

Also 6 times the total number of odd entries in first n rows of Pascal's triangle A007318, see formula section.

The structure contains three distinct kinds of polygons formed by triangular ON cells: the initial hexagon, rhombuses (each one formed by two ON cells) and unit triangles.

Note that if n is a power of 2 greater than 2, the structure looks like concentric hexagons with triangular holes, where some of them form concentric stars.